Semi–Lagrangian Approximation Schemes for Linear and Hamilton ...

✐✐Semi–Lagrangian Approximation Schemesfor Linear and Hamilton–Jacobi EquationsMaurizio Falcone and Roberto FerrettiJune 6, 2012

✐✐iiWARNINGThis is a draft NOT intended for free circulation. It contains the first six chapters ofthe book which will be published by SIAM. Three more chapters deal with controland game problems, front propagation and fluid dynamics.If you want to use this material, please contact the authors via email:Maurizio Falcone, falcone@mat.uniroma1.itRoberto Ferretti, ferretti@mat.uniroma3.itRome, February 2, 2011

✐✐ContentsPrefaceiii1 Models and motivations 11.1 Linear advection equation . . . . . . . . . . . . . . . . . . . . . 11.2 Nonlinear evolutive problems of Hamilton–Jacobi type . . . . . 31.3 Nonlinear stationary problems . . . . . . . . . . . . . . . . . . . 61.4 Simple examples of approximation schemes . . . . . . . . . . . . 111.5 Some difficulties arising in the analysis and in the approximation 141.6 How to use this book . . . . . . . . . . . . . . . . . . . . . . . . 152 Viscosity Solutions of First Order PDEs 172.1 The definition of viscosity solution . . . . . . . . . . . . . . . . . 172.1.1 Some properties of viscosity solutions . . . . . . . . 192.1.2 An alternative definition of viscosity solution . . . . 232.1.3 Uniqueness of viscosity solutions . . . . . . . . . . . 262.2 Viscosity solution for evolutive equations . . . . . . . . . . . . . 272.2.1 Representation formulae and Legendre transform . . 282.2.2 Semiconcavity and regularity of viscosity solutions . 322.3 Problems in bounded domains . . . . . . . . . . . . . . . . . . . 352.3.1 Boundary Conditions in a weak sense . . . . . . . . 362.4 Viscosity solutions and entropy solutions . . . . . . . . . . . . . 382.5 Discontinuous viscosity solutions . . . . . . . . . . . . . . . . . . 402.6 Commented references . . . . . . . . . . . . . . . . . . . . . . . 413 Elementary building blocks 433.1 A review of ODE approximation schemes . . . . . . . . . . . . . 433.1.1 One-step schemes . . . . . . . . . . . . . . . . . . . 443.1.2 Multistep schemes . . . . . . . . . . . . . . . . . . . 453.2 Reconstruction techniques in one and multiple space dimensions 473.2.1 Symmetric Lagrange interpolation in R 1 . . . . . . . 483.2.2 Essentially Non-Oscillatory interpolation in R 1 . . . 523.2.3 Weighted Essentially Non-Oscillatory interpolationin R 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.4 Handling multiple dimensions by separation of variables. . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2.5 Finite element interpolation . . . . . . . . . . . . . . 643.3 Function minimization . . . . . . . . . . . . . . . . . . . . . . . 693.3.1 Direct search methods . . . . . . . . . . . . . . . . . 70v

✐✐viContents3.3.2 Descent methods . . . . . . . . . . . . . . . . . . . . 713.3.3 Powell’s method and its modifications . . . . . . . . 723.3.4 Trust-region methods based on quadratic models . . 733.4 Numerical computation of the Legendre transform . . . . . . . . 733.5 Commented references . . . . . . . . . . . . . . . . . . . . . . . 754 Convergence theory 774.1 The general setting . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Convergence results for linear problems: the Lax–Richtmeyertheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . 794.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.3 Convergence . . . . . . . . . . . . . . . . . . . . . . 814.2.4 Time-dependent evolution operators . . . . . . . . . 824.2.5 The stationary case . . . . . . . . . . . . . . . . . . 834.3 More on stability . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.1 The CFL condition . . . . . . . . . . . . . . . . . . 854.3.2 Monotonicity and L ∞ stability . . . . . . . . . . . . 864.3.3 Monotonicity and Lipschitz stability . . . . . . . . . 884.3.4 Von Neumann analysis and L 2 stability . . . . . . . 894.4 Convergence results for Hamilton–Jacobi equations . . . . . . . 914.4.1 Crandall–Lions and Barles–Souganidis theorems . . 914.4.2 Lin–Tadmor theorem and semi–concave stability . . 984.5 Numerical diffusion and dispersion . . . . . . . . . . . . . . . . . 1004.6 Commented references . . . . . . . . . . . . . . . . . . . . . . . 1025 First-order approximation schemes 1035.1 Treating the advection equation . . . . . . . . . . . . . . . . . . 1035.1.1 Upwind discretization: the First Order Upwind scheme1045.1.2 Central discretization: the Lax–Friedrichs scheme . 1085.1.3 Semi–Lagrangian discretization: the Courant–Isaacson–Rees scheme . . . . . . . . . . . . . . . . . . . . . . 1135.1.4 Multiple space dimensions . . . . . . . . . . . . . . . 1235.1.5 Boundary conditions . . . . . . . . . . . . . . . . . . 1255.1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . 1265.2 Treating the convex HJ equation . . . . . . . . . . . . . . . . . . 1285.2.1 Upwind discretization . . . . . . . . . . . . . . . . . 1295.2.2 Central discretization . . . . . . . . . . . . . . . . . 1315.2.3 Semi–Lagrangian discretization . . . . . . . . . . . . 1335.2.4 Multiple space dimensions . . . . . . . . . . . . . . . 1385.2.5 Boundary conditions . . . . . . . . . . . . . . . . . . 1395.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.4 Stationary problems . . . . . . . . . . . . . . . . . . . . . . . . . 1415.4.1 The linear case . . . . . . . . . . . . . . . . . . . . . 1425.4.2 The nonlinear case . . . . . . . . . . . . . . . . . . . 1455.5 Commented references . . . . . . . . . . . . . . . . . . . . . . . 1466 High-order SL approximation schemes 1496.1 Semi-Lagrangian schemes for the advection equation . . . . . . 1496.1.1 Construction of the scheme . . . . . . . . . . . . . . 149

✐✐viiiContents8.3.2 Treatment of diffusive terms . . . . . . . . . . . . . 2498.4 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 2498.4.1 Linear advection . . . . . . . . . . . . . . . . . . . . 2498.4.2 Examples in R . . . . . . . . . . . . . . . . . . . . . 2498.4.3 Examples in R 2 . . . . . . . . . . . . . . . . . . . . . 2498.5 Commented references . . . . . . . . . . . . . . . . . . . . . . . 250Bibliography 251Index 265

✐✐ContentsixBasic Notationsa.e. almost everywhere (with respect to the Lebesgue measure)R d euclidean d-dimensional spacex · y scalar product ∑ di=1 x iy i of two vectors x, y ∈ R d|x| euclidean norm of x ∈ R d ,|x| = (x · x) 1/2B(x 0 , r) open ball of center x 0 and radius r in R d , {x : |x − x 0 | < r}B(x 0 , r) closed ball of center x 0 and radius r in R d , {x : |x − x 0 | ≤ r}∂E boundary of the set Eint E interior of the set EE closure of the set Eco E convex hull of the set Ea ∨ b min{a, b}, for a, b ∈ Ra ∧ b max{a, b}, for a, b ∈ RDu(x) gradient of a function u : R d → R at the point xd(x, C) distance from the point x to the closed set C∆t, ∆x discretization steps,⎧in compact form ∆ ≡ (∆x, ∆t)⎪⎨ +1 if x > 0sgn(x) sign of x: sgn(x) := 0 if x = 0⎪⎩−1 if x < 0i imaginary unit{1 if x ∈ Ω1 Ω (x) characteristic function of the set Ω: 1 Ω (x) :=0 if x ∉ Ω{1 if i = jδ ij Kronecker’s symbol: δ ij :=0 if i ≠ j⌊x⌋ lower integer part of x: ⌊x⌋ := max{m ∈ Z : m ≤ x}I k [G](x) value at the point x of the interpolate of degree k of the function gbased on the vector G = (g(x i )) of grid values of gν(x) exterior normal to a surface at the point xy(x, t; s) the position at time s of the trajectory starting at x at time ty x,t (s; α) the position at time s of the trajectory starting at x at time tand driven by the control αy x (s; α) the position at time s of the trajectory starting at x at time t = 0and driven by the control αT V (w) total variation of the univariate function w: T V (w) = ∫ D |w′ (x)| dxω(δ) a modulus of continuityBUC(Ω) space of bounded uniformly continuous functions on the open set ΩC k (Ω) space of functions u : Ω → R with continuousk-th derivative on a domain ΩC0 k (Ω) space of functions with compact support belonging to C k (Ω)L p (Ω) space of functions u : Ω → R such that ∫ Ω up (x) dx < +∞, 1 ≤ p < +∞L ∞ (Ω) space of functions u : Ω → R such that ess sup|u(x)| < +∞L p loc (Ω) space of functions u ∈ Lp (K) for any compact set K ⊂ Ω, 1 ≤ p ≤ +∞‖u‖ p norm of u in L p (Ω), 1 ≤ p ≤ ∞W m,p (Ω) Sobolev space of functions u : Ω → R with the first m derivatives in L p (Ω)W k,ploc (Ω) Sobolev space of functions u with the first m derivatives in Lp (K)for any compact set K ∈ Ω

✐✐xContentsspace of p-th power summable vectors or sequencesspace of bounded vectors or sequencesV , v j numerical solution, as a vector and as a value at the node x jV n , vj n numerical solution, as a vector and as a value at the node (x j , t n )V (k) , v (k)j numerical solution at the iteration k, as a vector and as a valueat the node x j‖V ‖ p norm in l⎧p , renormalized if referred to a numerical solution:⎨∑ ) 1/α(∆x‖V ‖ α := 1 · · · ∆x d j |v j| α if α < ∞⎩max j |v j | if α = ∞.B t transpose of a matrix Bρ(B) spectral radius of a square matrix B ∈ R m×m‖B‖ p for a (possibly infinite) matrix B, natural operator norm in l p :‖BV ‖ p‖B‖ p := supV ‖V ‖ pl pl ∞‖B‖ ∞ ∞-norm of a matrix B: ‖B‖ ∞ = max i∑j |b ij|‖B‖ 1 1-norm of a matrix B: ‖B‖ 1 = max j∑i |b ij|‖B‖ 2 2-norm of a matrix B: ‖B‖ 2 = √ ρ(B t B)w, w lower and upper semicontinuous envelopes:{w(x) := lim inf y→x w(y)w(x) := lim sup y→x w(y)

✐✐Chapter 1Models and motivationsThis book is devoted to the study of some class of hyperbolic Partial DifferentialEquations, and to their approximation of Semi-Lagrangian type. Throughout thebook, we will try to keep a mathematically rigorous analysis as much as possible;however, before getting into the most technical details and results, we would liketo start by presenting some examples and applications which have motivated muchof the research in this area. In fact, the analysis and approximation of first orderPartial Differential Equations of Hamilton–Jacobi type has been driven by theirapplication to fields such as fluid dynamics, optimal control and differential games,image processing and material science just to mention the main ones.In all the aforementioned fields, the notion of weak solutions has had a crucialimpact in giving a sound theoretical framework for both the analytical and thenumerical study. The goal of this chapter is precisely to introduce the reader tothis theory via some model problems. We will start from the linear advectionequation, then turn to nonlinear models, either the evolutive ones arising in frontpropagation and optimal control, or the stationary ones including, e.g. the eikonalequation of the classical Shape-from-Shading problem in image processing. We willalso discuss some difficulties related to the numerical approximation of such models,and conclude the chapter by giving some hints on how to use the material presentedin the book.1.1 Linear advection equationAs a first model, we start by considering the advection equation:u t + f(x, t) · Du = g(x, t) (x, t) ∈ R d × (t 0 , T ). (1.1)Here, f : R d × (t 0 , T ) → R d is a vectorfield termed as drift and g : R d × (t 0 , T ) → Ris the source term. We look for a solution u : R d × (t 0 , T ) → R of (1.1) satisfyingthe initial conditionu(x, t 0 ) = u 0 (x) x ∈ R d . (1.2)This is a classical model describing the transport of a scalar field, e.g. the distributionof a pollutant emitted by one of more sources (represented by g), andtransported by a water stream or a wind (represented by f).This physical interpretation is better clarified through the method of characteristicswhich provides an alternative characterization of the solution. In the1

✐✐2 Chapter 1. Models and motivationssimplest case, for g(x, t) ≡ 0 and f(x, t) ≡ c (constant) the solution v is given bythe representation formulau(x, t) = u 0 (x − c(t − t 0 )) (x, t) ∈ R d × [t 0 , T ). (1.3)In fact, assume that a regular solution u exists. Deriving u with respect to t alonga curve of the form (y(t), t) we obtaindudt (y(t), t) = Du(y(t), t) · ẏ(t) + u t(y(t), t), (1.4)so that (using the fact that u solves the advection equation) the total derivativewill identically vanish along the curves which satisfyẏ(t) = c. (1.5)Such curves have the physical meaning of flow lines along which the scalar field uis transported. They are known as characteristics and, in this particular case, arestraight lines. Then, to assign a value to the solution at a point (x, t) it sufficesto follow the unique line passing through (x, t) until it crosses the x-axis at thepoint z = x − c(t − t 0 ), z being the foot of the characteristic. Since the solutionu is constant along this line, we get (1.3). Conversely, it is easy to check that afunction of x and t in the form (1.3) satisfies (1.1) for f ≡ c and g ≡ 0, provided u 0is differentiable.In the case of variable coefficients f and g in (1.1), we have the more generalform∫ tu(x, t) = u 0 (y(x, t; t 0 )) + g(y(x, t; s), s)ds, (1.6)t 0where y(x, t; s) is the (backward) solution at time s of the Cauchy problem, calledsystem of base characteristics{ẏ(x, t; s) = f(y(x, t; s), s)(1.7)y(x, t; t) = x.In this more general formula, which will be proved in Chapter 2, the value u(x, t)depends on the value at the foot of the characteristic curve (which might no longerbe a straight line), as well as on the integral of the source term along the characteristicitself. Again, it can be easily shown that a function defined by (1.6)–(1.7)solves (1.1)–(1.2) if u 0 is smooth.The representation formula (1.6)–(1.7) shows that the regularity of the initial conditionis preserved along the characteristics if both drift and source terms aresmooth enough, so that u 0 ∈ C p (R d ) and f, g ∈ C p−1 (R d × (t 0 , T )) imply thatu ∈ C p (R d × (t 0 , T )). A key point in this respect is that characteristics cannotcross, a property which only holds in the linear case.A typical result of existence and uniqueness of classical solutions to the advectionequation in one space dimension is the following.Theorem 1.1. Consider equation (1.1)–(1.2) and assume that f, g ∈ C 1 (R ×(t 0 , T )) and that u 0 ∈ C 1 (R×(t 0 , T )). Then, there exists a unique classical solutionof (1.1), i.e. a function u satisfying (1.1) at every point (x, t) ∈ R × (t 0 , T ).

✐✐1.2. Nonlinear evolutive problems of Hamilton–Jacobi type 3The proof of this result is based on the method of characteristics, which willbe better examined in Chapter 2 (a more general result in dimension d can befound in [E98]). A crucial point, beside the regularity of the coefficients and of theinitial condition, is that the field of characteristics (1, f(x, t)) should be transversalto the x-axis at t 0 so that the solution can be defined by the initial condition ona neighborhood of the x-axis. In practice, this correspond to the request that fshould not vanish over an interval.At a closer look, however, well-posedness results requiring differentiability ofthe initial condition might not be satisfactory in a number of applications whichnaturally lead to a nonsmooth definition of u 0 . For example, if we try to simulatethe advection of a pollutant spread over a precise area ω p , it would be natural toconsider a discontinuous initial condition such as{1 if x ∈ ω pu 0 (x) = 1 ωp =(1.8)0 else.Although we expect the physically relevant solution to be given again by (1.6)–(1.7), the lack of regularity in the initial condition does not enable the same formalderivation of the representation formula. Hence, it becomes necessary to introducea notion of weak solution which could allow to treat the Cauchy problem in a moregeneral setting. One way to achieve this goal is to look for weak solutions in thesense of distributions:Definition 1.2. A function u : R × (t 0 , T ) → R in L 1 (R × (t 0 , T )) is a weaksolution of the Cauchy problem (1.1)–(1.2) if and only if for any test function φ ∈C0(R 1 × (t 0 , T )) the following integral condition is satisfied:∫ Tt 0∫Ru(x, t)φ t (x, t) + [u(x, t)f(x, t)] φ x (x, t)dx dt = (1.9)=∫ Tt 0∫Rg(x, t)ϕ(x, t) dx dtIn recent years, this strategy of definition for weak solutions has also beenextended to include bounded variation solutions. Anyway, we will not pursue thisline of work here, and rather focus on the concept of viscosity solution, which hasproved to be well suited for the case of Hamilton–Jacobi equations.1.2 Nonlinear evolutive problems of Hamilton–JacobitypeAmong all extensions to nonlinear equations, this book will mostly address itself tothe following Hamilton–Jacobi equation:{u t + H(x, Du) = 0 (x, t) ∈ R d × (0, T )v(x, 0) = v 0 x ∈ R d (1.10),where H : R d → R is called the Hamiltonian of the equation and is typicallyassumed to be convex. We soon sketch a couple of applications which fall in thisclass of problems.

✐✐4 Chapter 1. Models and motivationsFigure 1.1. A representation function for Γ 0Front propagation via the level set method A special case of the Hamilton–Jacobiequation arise in modeling a front which evolves in the normal direction with a givenvelocity c : R d → R. The level set method considers the Cauchy problem{u t + c(x)|Du| = 0 (x, t) ∈ R d × (0, T )u(x, 0) = u 0 , x ∈ R d (1.11)where u 0 : R d → R must be a proper representation of the front Γ 0 , i.e. a continuousfunction u 0 which changes sign on Γ 0 := ∂Ω. In two space dimensions, this wouldlead to the definition:Definition 1.3. Assume that Γ 0 is a closed curve in R 2 and denote by Ω 0 the regionenclosed by Γ 0 . A continuous function u 0 : R 2 → R is a proper representation ofΓ 0 if and only if it satisfies the following conditions:⎧⎪⎨ u 0 (x) < 0 x ∈ Ω 0u 0 (x) = 0 x ∈ Γ 0(1.12)⎪⎩u 0 (x) > 0 x ∈ R 2 \ Ω 0In this way, the front Γ 0 at the initial time is identified with the zero-level setof u 0 . Assume now that we let Γ 0 evolve in the normal direction with velocity c(x).To fix ideas, we can assume that the normal η is directed outwards with respect toΩ and that the velocity is positive (this situation corresponds to an expansion ofthe front, whereas an opposite sign would produce a shrinking). Moreover, let usconsider the evolution of a curve in the (x, y)-plane to simplify the presention.The front at time t will be denoted by Γ t and a generic point P ∈ Γ t will be representedas P = (x 1 (s, t), x 2 (s, t)), where s refers to the parametric representation ofΓ t (e.g. as an arc length parametrization). Since the velocity V at every point hasmagnitude c(P ) and is directed in the normal direction η, we can writeV (P ) = (ẋ 1 (s, t), ẋ 2 (s, t)) = c(P )η(P ) (1.13)

✐✐1.2. Nonlinear evolutive problems of Hamilton–Jacobi type 5Figure 1.2. The propagation of a front in the normal directionAfter describing Γ 0 as the zero-level set of u 0 , our plan is to describe the front Γ tat a generic time t as the zero-level set of the solution of a Cauchy problem, havingu 0 as its initial condition. In order to derive the equation characterizing Γ t , weconsider the total derivative of u(x 1 (s, t), x 2 (s, t)). Taking into account that thisderivative vanishes on a level set of constant value, we have:0 = d dt u(x 1(s, t), x 2 (s, t), t) == u t (x 1 (s, t), x 2 (s, t)) + Du(x 1 (s, t), x 2 (s, t)) · (ẋ 1 , ẋ 2 ) == u t (P ) + c(P )Du(P ) · η(P ) == u t (P ) + c(P )|Du(P )|where the normal direction to a level set of u (computed at P ) has been writtenin terms of u as Du(P )/|Du(P )|. In conclusion, solving (1.11) and looking at thezero-level set of the solution u(x, t) we can recover the front Γ t . Note that the choiceof the zero level is arbitrary, and in fact all level sets of u are moving according tothe same law.Once formulated the front evolution by means of (1.11), we can drop thesmoothness assumption on Γ t , and apply the same method to more general initialconfigurations. Here, the possibility of defining weak (nondifferentiable) solutionsfor (1.11) plays a crucial role. For example, taking Γ 0 as the union of a numberof separate curves in the plane, if the evolution is driven by a positive velocitythe various components of the front will collide in a finite time and we will have amerging of the fronts. It is clear that, at a point where two components of the frontare tangent, we cannot expect to have a normal vector and a singularity appears inthe evolution (and in the equation). However, as we will see later on, the level setmethod relies on a weak concept of solution which is robust enough to handle boththe onset of a singularity and the change of topology.This quick explanation of the level set method also shows its main drawback.In order to follow a (d − 1)-dimensional front we are forced to embed the problemin R d , and this implies an extra computational cost. This remark has motivated

✐✐6 Chapter 1. Models and motivationsthe development of “local” versions of the level set method which reduce the computationalcost by taking into account only the values in a small neighbourhood ofthe front, the so-called narrow band. More details on such methods will be given inChapter ??.Optimal control and dynamic programming A second relevant application of theHamilton–Jacobi equation is to characterize the value function of a finite horizonoptimal control problem. Let a controlled dynamical system be described by{ẏ(t) = b(y(t), t, α(t))(1.14)y(t 0 ) = x 0where y ∈ R d is the state of the system, α : (t 0 , T ) → A ⊆ R m is the controlfunction, b : R d × (t 0 , T ) × A → R d is the controlled drift (or dynamics).Let s consider the class of admissible controls A := {α : (t 0 , T ) → A, measurable}where A is a compact subset of R m since it is known that for any given measurablecontrol α(·) the solution of (1.14) is well defined (in a weak sense). Then we canintroduce the cost functional J : A → R which will be used to select the “optimal”(i.e., the minimizing) trajectory. For the finite horizon problem, the functional isusually defined asJ(x, t 0 ; α(·)) =∫ Tt 0g(y(s), s, α(s))e −λ(s−t0) ds + e −λ(T −t0) g(y(T )) λ > 0(1.15)where g : R d × (t 0 , T ) × A → R is the running cost depending on the state of thesystem, on the time and on the control, λ ∈ R + is the discount factor which allowsto compare costs at different times by rescaling them at time t 0 and g is a terminalcost. The value function associated to this problem is defined asu(x, t) = inf J(x, t; α(·)) (1.16)α(·)The value function represents the best value we can get from our cost functional and,as it will be shown in Chapter 7, enables to construct the optimal solution. In allclassical deterministic control problems, the Dynamic Programming principle allowsto characterize the value function as the unique solution, once more in a suitablyweak sense, of a partial differential equation known as the Dynamic Programming(or Bellman) equation:⎧⎨u t + sup[−b(x, t, a) · Du − g(x, t, a)] = 0 (x, t) ∈ R d × (0, T )a∈A(1.17)⎩u(x, T ) = g(x) x ∈ R d ,where a ∈ A is a parameter spanning the control set. Note that, unless for being abackward equation, (1.17) is in the Hamilton–Jacobi form (1.10). More details onthis technique will be given in Chapter 7, whereas a more extensive presentationcan be found in [BCD97].1.3 Nonlinear stationary problemsWe turn now to some examples of stationary Hamilton–Jacobi equations which willbe analyzed in the sequel. Problems of this kind may arise when dealing withapplications related to control theory and image processing.

✐✐8 Chapter 1. Models and motivationsFigure 1.3. A case in which d(·, Ω) is not differentiablewhere α : (0, +∞) → B(0, 1) is a measurable control. Given a closed convex targetset Ω, we want to find the minimal time necessary to drive the system from itsinitial position to the target (the minimal time represents the value function ofthis particular problem). Since the system can move in every direction, the optimaltrajectory is a straight line connecting the initial position x 0 to its unique projectionon Ω. Moreover, the velocity has at most unit norm, and therefore the minimumtime of arrival from x 0 equals the distance d(x 0 , Ω) and solves the eikonal equation(1.19).More in general (as we will see in Chapter 7 devoted to control problems andgames), if the system is driven by a nonlinear controlled dynamical system of theform {ẏ(t) = b(y(t), α(t))y(0) = x 0 ∈ R d (1.25)\ Ω,we end up with the following Bellman equation for the minimum time function:{max [−b(x, a) · Du] = 1 xa∈Au(x) = 0∈ Rd \ Ωx ∈ ∂Ω.(1.26)The characterization of the value function in terms of a stationary Bellmanequation also applies to other deterministic optimal control problems. For example,we mention that an equation of the formλu(x) + max[−b(x, a) · Du(x) − g(x, a)] = 0 (1.27)a∈Acharacterizes the value function of an infinite horizon problem for the system (1.25),i.e. the functionu(x) := inf J(x, α(·)), (1.28)α∈Awith a cost functional J(x, α(·)) given byJ(x, α) :=∫ +∞0g(y(s), α(s))e −λs ds. (1.29)

✐✐1.3. Nonlinear stationary problems 9Figure 1.4. Optimal trajectories of the minimum time problem for a squaretarget Ω.Figure 1.5. A gray-level picture (left) and the corresponding surface (right).We refer again to Chapter 7 for a more detailed presentation of Dynamic Programmingtechniques.Shape-from-Shading An equation of eikonal type also arises in the Shape-from-Shading problem, a classical inverse problem in which one tries to reconstruct, usingthe shading information, the surface which has generated a certain monochromaticpicture taken by a camera.To be more precise, we introduce the basic assumptions and notations for thisproblem. We attach to the camera a three-dimensional coordinate system (Oxyz),such that Oxy coincides with the image plane and Oz coincides with the optical

✐✐10 Chapter 1. Models and motivationsaxis. Under the assumption of orthographic projection, the visible part of the sceneis, up to a scale factor, a graph z = u(x), where x = (x 1 , x 2 ) is an image point.Horn has shown in [HB89] that the Shape-from-Shading problem can be modeledby the image irradiance equation:R(ν(x)) = I(x), (1.30)where I(x) is the grey level measured in the image at point x (in fact, I(x) is theirradiance at point x, but the two quantities are proportional) and R(ν(x)) is thereflectance function, giving the value of the light re-emitted by the surface as afunction of its orientation, i.e., of the unit normal ν(x) to the surface at a point(x, u(x)). For the graph of a function u, this normal can be expressed as:ν(x) =1√ (−p(x), −q(x), 1), (1.31)1 + p(x)2 + q(x)2where p = ∂u/∂x 1 and q = ∂u/∂x 2 , so that Du(x) = (p(x), q(x)). The irradiancefunction (or image brightness) I is given as a measure of grey level (from 0 to 255in digital images) at each pixel of the image, and in order to construct a continuousmodel, we will assume that it is renormalized in the interval [0, 1]. We also assumethat there is a unique light source at infinity, whose direction (supposed to beknown) is given by the unit vector ω = (ω 1 , ω 2 , ω 3 ) ∈ R 3 .The height function u, which is the unknown of the problem, has to be reconstructedon a compact domain Ω ⊂ R 2 , called the reconstruction domain. Recallingthat, for a Lambertian surface of uniform unit albedo, R(ν(x)) = ω ·ν(x), and using(1.31), then (1.30) can be rewritten for any x ∈ Ω asI(x) √ 1 + |Du(x)| 2 + (ω 1 , ω 2 ) · Du(x) − ω 3 = 0, (1.32)which is a stationary first-order equation of Hamilton–Jacobi type. Points x ∈ Ωsuch that I(x) is maximal correspond to the particular situation where ω and ν(x)point in the same direction: these points are usually called “singular points”. Themost widely studied case in the Shape-from-Shading literature corresponds to afrontal light source at infinity, i.e., ω = (0, 0, 1). Then, (1.32) becomes an eikonalequation of the form:√1 − I|Du(x)| =2 (x)I 2 (1.33)(x)It is known that the eikonal equation may not have a unique solution wheneverthe right-hand side vanishes at some point. So the existence of singular pointscorresponds to an intrinsic ill-posedness of the Shape-from-Shading problem. Infact, a simple example can show that uniqueness does not hold even for classicalsolutions. Consider a one-dimensional case in which the surface is given by u 1 (x) =1 − x 2 in the interval [−1, 1]. This surface clearly satisfies the boundary conditionsu 1 (1) = u 1 (−1) = 0 and, assuming a vertical light source with ω = (0, 1) it alsosatisfies (1.33) for the corresponding irradiance, which is given byI(x) =(0, 1) · (2x, 1)√1 + 4x2(1.34)At the point x = 0, I takes its maximum value I(0) = 1, and therefore the righthandside of (1.33) vanishes. Note that the Lambertian assumption on the surface

✐✐1.4. Simple examples of approximation schemes 11Figure 1.6. One classical solution and some a.e. solutions of the Shapefrom-Shadingproblem.implies that I only depends on the angle between the normal ν(x) and ω, so thatit is clear that the mirror image of the surface u 1 with respect to the x-axis, i.e.u 2 (x) = −1 + x 2 is still a classical solution of (1.33). It is natural to observe (seeFigure 1.6) that all mirror reflections of a part of both surfaces u 1 and u 2 withrespect to a horizontal line satisfy (1.33) almost everywhere (although this does notimply that they could be acceptable weak solutions).1.4 Simple examples of approximation schemesIn this section, we try to give some very basic ideas (in a single space dimension)about the approximation schemes for the model problems introduced so far. Westart with the case of the rigid transport problem, i.e. the linear advection atconstant speed c (to fix ideas, we assume that c > 0), which is described by theequation {u t + cu x = 0 (x, t) ∈ R × (0, T )(1.35)u(x, 0) = u 0 (x) x ∈ R.The most classical method to construct an approximation of (1.35) is to build auniform grid in space and time (a lattice) with constant steps ∆x and ∆t, coveringthe domain of the solution:{(x j , t n ) = (j∆x, n∆t), j ∈ Z, n ∈ N, n ≤ T }. (1.36)∆tThe basic idea behind all finite difference approximation is to replace every derivativeby an incremental ratio. Thus, one obtains a finite dimensional problem whoseunknown are the values of the numerical solution at all the nodes of the lattice,so that the value u n j associated to the node (x j, t n ) should be regarded as an approximationof u(x j , t n ). For the time derivative it is natural to choose the forwardincremental ratiou(x, t + ∆t) − u(x, t)u t (x, t) ≈ (1.37)∆twhich allows, starting with the solution at the initial time t 0 , to compute explicitlythe approximations for increasing times t k > t 0 . Writing at (x j , t n ) the forwardincremental ratio in time, we getu t (x j , t n ) ≈ un+1 j − u n j∆t(1.38)

✐✐12 Chapter 1. Models and motivationsFigure 1.7. Approximation of the rigid transport problem by the upwindscheme (left, ∆t = 0.025) and by the Semi-Lagrangian scheme (right, ∆t = 0.05).For the approximation of u x we have more choices, likeu x (x j , t n ) ≈ un j+1 − un j∆xu x (x j , t n ) ≈ un j − un j−1∆xu x (x j , t n ) ≈ un j+1 − un j−12∆x(right finite difference)(left finite difference)(centered finite difference),(1.39)which are all based on the values at the nodes (x k , t n ).In this case, the choice of the approximation for u x crucially affects the convergenceof the numerical solution to the exact solution. Although centered finite differencewould in principle provide a more accurate approximation of the space derivative,as we will see later on, given the time derivative approximation (1.38), the onlychoice in (1.39) which results in a convergent scheme is to use the left incrementalratio if c > 0, the right incremental ratio if c < 0, which corresponds to the so-calledupwind scheme. In conclusion, for c > 0, we obtain a scheme in the formu n+1j− u n j∆t+ c un j − un j−1∆x= 0. (1.40)A different way to construct the approximation of (1.35) is to consider theadvection term as a directional derivative and writecu x (x j , t n ) ≈ − un (x j − cδ) − u n jδ(1.41)where δ is a “small” positive parameter, and u n denotes an extension of the numericalsolution (at time t n ) to be computed outside of the grid. Coupling the forwardfinite difference in time with this approximation we getu n+1j− u n j∆t− un (x j − cδ) − u n jδand finally, choosing δ = ∆t, we obtain the scheme= 0,u n+1j = u n (x j − c∆t) (1.42)

✐✐14 Chapter 1. Models and motivations1.5 Some difficulties arising in the analysis and in theapproximationAfter giving models, applications and motivations for the theory which will be theobject of this book, it is worth to point out a number of mathematical peculiaritiesof the problems under consideration.1. The analysis of linear transport problems and Hamilton–Jacobi equation leadsto drop classical solutions and introduce weak solutions in order to treat relevantapplications. The analysis of weak solutions in this framework hasstarted in the 60s with the pioneering works by S.N. Kružkov and has proceededthrough the last decades of the century when M.G. Crandall and P.L.Lions introduced the notion of viscosity solution. Typically, viscosity solutionsare Lipschitz continuous but, at least for convex Hamiltonians, the conceptcan be extended to discontinuous solutions. Although the theory is still undergoingtechnical developments, and new classes of problems are analyzedin this framework, this book will mainly refer to the basic theory for convexHamiltonians, which is at the moment quite an established matter.2. The lack of smoothness of viscosity solutions makes it difficult to developefficient approximations. Starting from the 80s, monotone finite differencemethods have been proposed using the relationship between Hamilton–Jacobiequations and conservation laws (this relationship will be discussed in thenext chapter). On this basis, monotone finite difference methods conceived forconservation laws have been adapted to the approximation of Hamilton–Jacobiequation. However, monotone schemes are typically very diffusive, a seriousdrawback when treating nonsmooth problems. On the other hand, Semi-Lagrangian schemes may provide less diffusive, still monotone approximationsthrough the use of large time steps. Construction and convergence theory ofSemi-Lagrangian schemes, as well as their comparison with difference schemes,will be a major subject of this book.3. In the last decades, the low efficiency of monotone scheme has motivated alarge effort to develop less diffusive, high-order approximation schemes. Thisongoing research, which can have a great impact on applications, has alreadyproduced a number of methods of great accuracy, although very few theoreticalresults are available. In this book, we will try to synthesize the existing resultsfor high-order Semi-Lagrangian schemes into a rigorous convergence analysis,at least on model cases.4. The theory of viscosity solutions has been tested on a variety of applications,ranging from control and games theory to image processing, from fluid dynamicsto combustion and geophysics, and has proved to be particularly effectivein characterizing the relevant solution. Some chapters of this book will bedevoted to these topics. However, these applications often require to work inhigh dimension. Along with their theoretical study, the development of fastand accurate numerical schemes for high-dimensional problems seems to be acritical, challenging task for future research.

✐✐1.6. How to use this book 151.6 How to use this bookWe conclude this introductory chapter by giving, on the basis of our personal experience,some hints about the use of the material presented in the book. Such hintscan also be useful for a reader approaching the subject for the first time.• A basic introductory course on the analysis and approximation of Hamilton–Jacobi equation can use the first three chapters if you want to emphasize themathematical aspects.• An advanced numerical course on nonlinear hyperbolic differential equationswill include, for example, Chapters 2, 4, 3, 6.• A course on applications to front propagation, level set methods and fluiddynamics can use the material in Chapters 2, 3, 4, and 8 or 9 depending onthe emphasis to be given to different topics.• A course on applications to control theory can use the material in Chapters2, 3, 4 and 7.We will be glad to receive other suggestions for the use of this book for graduateand advanced undergraduate courses.

✐✐16 Chapter 1. Models and motivations

✐✐Chapter 2Viscosity Solutions ofFirst-Order PDEsThis chapter contains a general presentation of the theory of weak solutions in theviscosity sense for first-order partial differential equations, starting from the stationarycase (which include, e.g., the eikonal equation), and moving to evolutivemodels (which arise in control theory and front propagation).The chapter reviews basic and essential results of existence and uniqueness forviscosity solutions. Moreover, it also presents other related topics such as characteristicsand Hopf–Lax formulae, regularity results (typically, based on the notion ofsemiconcavity), boundary conditions, and the relationship with entropic solutionsof conservation laws. Most of these topics will prove useful in the construction andanalysis of the approximation schemes presented in the sequel. Finally, we brieflysketch the concept of discontinuous viscosity solutions.2.1 The definition of viscosity solutionTo introduce the theory of weak solutions in the viscosity sense, we start by consideringa stationary problem of the general formH(x, u, Du) = 0, x ∈ Ω. (2.1)Here, Ω is an open domain of R d and the Hamiltonian H : R d × R × R d → R is acontinuous real valued function on Ω×R×R d . In this first discussion, we will avoidto treat boundary conditions. In fact, boundary conditions need to be understoodin a suitable, more delicate “weak sense” which will be examined later on in thischapter.Typically, the main results in the theory of viscosity solutions assume that theHamiltonian H(x, u, p) satisfies the following basic set of assumptions:(A1) H(·, ·, ·) is uniformly continuous on Ω × R × R d(A2) H(x, u, ·) is convex on R d(A3) H(x, ·, p) is monotone on R.The need for a notion of weak solution arises when observing that problem (2.1) isnot expected to have a classical (i.e., C 1 (Ω)) solution, as the following counterexampleshows.17

✐✐18 Chapter 2. Viscosity Solutions of First Order PDEsA counterexampleConsider the following problem in one dimension:{|u x | = 1 x ∈ Ω ≡ (−1, 1)u(x) = 0 x ∈ ∂Ω.(2.2)Obviously, u 1 (x) = x and u 2 (x) = −x are C 1 and satisfy the equation pointwise in(−1, 1) but they do not satisfy both boundary conditions.On the other hand, if the assumption of differentiability is dropped, we mightconsider functions which satisfy the equation almost everywhere and satisfy theboundary conditions.Two such solutions would be given by the functions u 3 (x) = |x| − 1 and u 4 (x) =1 − |x|, but in fact, by collecting piecewise affine functions with slopes ±1 it ispossible to construct infinitely many a.e. solutions of the equation. Therefore, it isclear that the notion of “a.e. solution” gives too many solutions and is unsuitable fora uniqueness result. One possibility to recover uniqueness is to perform an ellipticregularization, i.e., to regularize the problem by adding a second order term −εu xx .Consider the second order problem−εu xx + |u x | = 1, x ∈ (0, 1) (2.3)with the homogeneous boundary condition u(−1) = 0, u(1) = 2. For every positiveε, this problem has a regular C 2 ((−1, 1)) solution u ε , which can be explicitly writtenasu ε (x) = x + exp ( ) ( )x−1ε − exp −1ε1 − exp ( )− 1 (2.4)εSince (2.2) is the limit problem of (2.3) as ε → 0, we would like to know thelimiting behaviour of u ε . In any compact set contained in (0, 1), u ε convergesuniformly to u(x) = x. The main trouble appears near the boundary since theboundary condition causes the occurrence of a boundary layer of width O( √ ε). Thesolution u ε must attain the value u(x) = 2 at x = 1, and in the limit this produces adiscontinuity. Assuming that the limiting process converges to a continuous functionu, we can pass to the limitlimε→0 uε = u (2.5)and define u to be the weak solution of our problem. The definition of viscosity solutionavoids the limiting procedure related to the vanishing viscosity approximationand gives a local characterization of the weak solution.The “classical” definition We first give the most popular definition of viscositysolution, for u ∈ BUC(Ω) (the space of Bounded Uniformly Continuous functionover the open set Ω).Definition 2.1. Let u ∈ BUC(Ω). We say that u is a viscosity solution of (2.1)if and only if, for any ϕ ∈ C 1 (Ω), the following conditions hold:(i) at every point x 0 ∈ Ω, local maximum for u − ϕ,(i.e., u is a viscosity subsolution);H(x 0 , u(x 0 ), Dϕ(x 0 )) ≤ 0

✐✐2.1. The definition of viscosity solution 19(ii) at every point x 0 ∈ Ω, local minimum for u − ϕ,(i.e., u is a viscosity supersolution).H(x 0 , u(x 0 ), Dϕ(x 0 )) ≥ 0Going back to the counterexample, we explain how such a definition allows toselect a unique a.e. solution.First, we show that any a.e. solution u which has a local minimum at x 0 cannot bea viscosity supersolution. In fact, choose ϕ ≡ c (a constant). Clearly, x 0 is a localminimum point for u − ϕ, and therefore we should have|ϕ x (x 0 )| ≥ 1which is false since ϕ x ≡ 0.Note that the same argument works in a different way for subsolutions. Take ana.e. solution u which has a local maximum at x 0 and choose ϕ ≡ c. Clearly, x 0 isa local maximum point for u − ϕ so that we should have|ϕ x (x 0 )| ≤ 1which is true, since ϕ x ≡ 0. This implies that the viscosity solution can havemaximum but not minimum points. The only viscosity solution of our problem istherefore u(x) = 1 − |x|.2.1.1 Some properties of viscosity solutionsWe review here some properties of viscosity solutions which will be useful in thesequel. In particular, we want to examine the link between classical and viscositysolutions, as well as some other properties which are relevant for the constructionof approximation schemes.Relationship with classical solutions The first result states the local character ofviscosity solutions, along with their relationship with classical solutions.Proposition 2.2. The following statements hold true:(i) If u ∈ C(Ω) is a viscosity solution of (2.1) in Ω, then u is a viscosity solutionof the same equation in any open set Ω ′ ⊂ Ω.(ii) If u is a classical C 1 (Ω) solution, then it is also a viscosity solution.(iii) Any viscosity solution that is also regular is a classical solution.Proof. (i) If x 0 is a local maximum point on Ω ′ for u − ϕ, ϕ ∈ C 1 (Ω ′ ) then x 0is a local maximum point on Ω for u − ˜ϕ, for any test function ˜ϕ ∈ C 1 (Ω) suchthat ˜ϕ ≡ ϕ on B(x 0 , r) for some positive r. Then, by the definition of viscositysubsolutions,H(x 0 , u(x 0 ), Dϕ(x 0 )) = H(x 0 , u(x 0 ), D ˜ϕ(x 0 )) ≤ 0.(ii) Assume u ∈ C 1 is a classical solution, so that H(x, u(x), Du(x)) = 0 for every

✐✐20 Chapter 2. Viscosity Solutions of First Order PDEsx ∈ Ω. Consider a test function ϕ and a local maximum point for u − ϕ. It is notrestrictive to choose ϕ(x 0 ) = u(x 0 ). Then, we haveD(u − ϕ)(x 0 ) = 0. (2.6)This implies that Dϕ(x 0 ) = Du(x 0 ) so u is viscosity subsolution. The proof that uis a supersolution is analogous.(iii) We can always choose ϕ = u so that every point is a local maximum and alocal minimum for u − ϕ. Then, for any x ∈ Ω both inequalities will be satisfied:H(x, u(x), Du(x)) ≤ 0,H(x, u(x), Du(x)) ≥ 0.This implies that the equation is satisfied pointwise.Min/Max of viscosity solutions Viscosity subsolutions (respectively, supersolutions)are stable with respect to the max (respectively, the min) operator. Introducing,for u, v ∈ C(Ω), the notations:we have the stability result:(u ∨ v)(x) = max{u(x), v(x)}(u ∧ v)(x) = min{u(x), v(x)},Proposition 2.3. The following statements hold true:(i) Let u, v ∈ C(Ω) be viscosity subsolutions of the stationary equation in (2.1).Then, u ∨ v is a viscosity subsolution.(ii) Let u, v ∈ C(Ω) be viscosity supersolutions of the stationary equation in (2.1).Then, u ∧ v is a viscosity supersolution.Proof. Let x 0 be a local maximum point for u ∨ v − ϕ where ϕ ∈ C 1 (Ω) is our testfunction. Without loss of generality, we can assume that (u ∨ v)(x 0 ) = u(x 0 ). Sincex 0 is local maximum point for u − ϕ, we haveH(x 0 , u(x 0 ), Dϕ(x 0 )) ≤ 0which proves (i). The reverse assertion (ii) can be proved is a similar way.An important property which follows from Proposition 2.3 is that the viscositysolution u can be characterized as the maximal subsolution of the equation, i.e.,u ≥ v, for any v ∈ S (2.7)where S is the space of subsolutions, i.e. S = {v ∈ C(Ω) : condition (i) is satisfied}.Proposition 2.4. Let u ∈ C(Ω) be a viscosity subsolution of the stationary equationin (2.1), such that u ≥ v for any viscosity subsolution v ∈ C(Ω). Then, u is aviscosity supersolution and therefore a viscosity solution of (2.1).

✐✐2.1. The definition of viscosity solution 21Proof. We will prove the result by contradiction. Assume thatd := H(x 0 , u(x 0 ), Dϕ(x 0 )) < 0for some ϕ ∈ C 1 (Ω) and x 0 ∈ Ω such thatu(x 0 ) − ϕ(x 0 ) ≤ u(x) − ϕ(x)∀x ∈ B(x 0 , δ 0 ) ⊂ Ωfor some δ 0 > 0. Now, consider the function w ∈ C 1 (Ω) defined asw(x) := ϕ(x) − |x − x 0 | 2 + u(x 0 ) − ϕ(x 0 ) + 1 2 δ2for 0 < δ < δ 0 . It is easy to check that, by construction,(u − w)(x 0 ) < (u − w)(x) ∀x such that |x − x 0 | = δ. (2.8)We prove now that, for some sufficiently small δ,H(x, w(x), Dw(x)) ≤ 0 ∀x ∈ B(x 0 , δ). (2.9)For this purpose, a local uniform continuity argument shows that, for 0 < δ < δ 0 ,{|ϕ(x) − ϕ(x 0 )| ≤ ω 1 (δ),|Dϕ(x) − 2(x − x 0 ) − Dϕ(x 0 )| ≤ ω 2 (δ) + 2δ(2.10)for any x ∈ B(x 0 , δ), where the ω i , i = 1, 2, are the moduli of continuity of respectivelyϕ and Dϕ. Then, we haveNow,|w(x) − u(x 0 )| ≤ ω 1 (δ) + δ 2 x ∈ B(x 0 , δ)H(x, w(x), Dw(x)) = d + H(x, w(x), Dw(x)) = (2.11)= d + H(x, w(x), Dϕ(x) − 2(x − x 0 )) − H(x 0 , w(x 0 ), Dϕ(x 0 )).Denoting by ω the modulus of continuity of H, we can writeH(x, w(x), Dw(x)) ≤ d + ω(δ, ω 1 (δ) + δ 2 , ω 2 (δ) + 2δ),for all x ∈ B(x 0 , δ). Since d is negative, the above inequality proves (2.9) for δ > 0small enough. Let us fix such a δ, and set̂v(x) :={u ∨ w on B(x 0 , δ)u on Ω \ B(x 0 , δ).(2.12)It is easy to check that, by (2.8), ̂v ∈ C(Ω), so by Propositions (2.2) and (2.3) (i) ̂vis a subsolution of (2.1). Since ̂v(x 0 ) > u(x 0 ) the statement is proved.

✐✐22 Chapter 2. Viscosity Solutions of First Order PDEsUniform convergence of viscosity solutions Viscosity solutions are also stablewith respect to uniform convergence in C(Ω), as the following result shows. It isworth to point out that this property does not hold for generalized a.e. solutions.Proposition 2.5. Let u n ∈ C(Ω), n ∈ N, be a viscosity solution ofH n (x, u n (x), Du n (x)) = 0 x ∈ Ω. (2.13)Assume thatu n → u locally uniformly in ΩH n → H locally uniformly in Ω × R × R d .Then, u is a viscosity solution of (2.1).(2.14)Proof. Let ϕ ∈ C 1 (Ω) and x 0 be a local maximum point for u − ϕ. It is notrestrictive to assume thatu(x 0 ) − ϕ(x 0 ) > u(x) − ϕ(x)for x ≠ x 0 , x ∈ B(x 0 , r) for some r > 0. By uniform convergence, and for n largeenough, u n − ϕ attains a local maximum at a point x n close to x 0 . Then, we haveH n (x n , u n (x n ), Dϕ n (x n )) ≤ 0.Since x n tends to x 0 for n → +∞, passing to the limit we getH(x n , u(x 0 ), Dϕ(x 0 )) ≤ 0The proof that u is also a viscosity supersolution can be obtained by a similarargument.In addition, the viscosity solution u may be characterized as the uniform limit(for ε → 0) of classical solutions u ε of the regularized problem−εu ε xx + H(x, u ε , Du ε ) = 0, (2.15)that is,lim uε = u.ε→0 +This explains the name of viscosity solutions, since the term −εu xx corresponds tothe viscosity term in fluid dynamics. In principle, for a given ε, the regularizationterm −εu ε xx allows to use the theory of elliptic equations to establish the existenceof a regular solution u ε ∈ C 2 (Ω). Nevertheless, passing to the limit for ε → 0 isquite a technical point, since the regularizing effect of the second order term is cutdown as ε vanishes. Moreover, for what we have seen in this chapter, we do notexpect to have a regular solution for the limiting (first order) equation. Withoutproving a precise result (which can be found in [Ba93]), we try give an idea of therelationship between the definition of viscosity solution and the limiting behaviourof u ε .Assume that u ε ∈ C 2 (R) converge uniformly as ε → 0 to some u ∈ C(R). Letus now take a regular function ϕ ∈ C 2 (R) and assume that x is a local maximumpoint for u − ϕ. By uniform convergence, u ε − ϕ attains a local maximum at somepoint x ε andlim xε = x.ε→0 +

✐✐2.1. The definition of viscosity solution 23By the maximum properties, we getD(u ε − ϕ)(x ε ) = 0, −D 2 (u ε − ϕ)(x ε ) ≥ 0where D 2 denotes the second derivative operator. Then, using the previous inequalitiesand the fact that u ε is a classical solution of (2.15), we obtain−εϕ xx (x ε ) + H(x ε , u(x ε ), Dϕ(x ε )) ≤ (2.16)−εu ε xx(x ε ) + H(x ε , u(x ε ), Du(x ε )) = 0Passing to the limit and using the continuity of ϕ xx , Dϕ and H we finally getH(x, u(x), Dϕ(x)) ≤ 0.This computation explains why, from the theoretical point of view, it is reasonableto obtain viscosity solutions via an elliptic regularization of the first order equation.Such a technique, however, is not as convenient for a numerical approach, sincethe introduction of a second-order term has a smoothing action. In particular, thiseffect is apparent on the singularities of solutions, which are definitely a point ofinterest in the approximation of Hamilton–Jacobi equations.Before concluding the section, we give a result concerning changes of variablesin (2.1). In the sequel, this result will be useful to deal with certain control problems.Proposition 2.6. Let u ∈ C(Ω) be a viscosity solution of (2.1) and Φ : R → R bea function in C 1 (R) such that Φ ′ (t) > 0. Then v := Φ(u) is a viscosity solution ofH ( x, Φ −1 ((v(x)), Φ −1 (v(x))Dv(x) ) = 0 x ∈ Ω. (2.17)We conclude this review with a warning about the fact that viscosity solutionsare not preserved by a change of sign in the equation. This behaviour, althoughunusual, is easily understood if we go back to the definition, which only takes intoaccount the local properties at minimum and maximum points for u−ϕ. Indeed, thefact that any local maximum of u−ϕ is a local minimum of −(u−ϕ) implies that uis a viscosity subsolution of (2.1) if and only if v = −u is viscosity supersolution of−H(x, −v(x), −Dv(x)) = 0 in Ω. Similarly, u is a viscosity supersolution of (2.1)if and only if v = −u is a viscosity subsolution of −H(x, −v(x), −Dv(x)) = 0 in Ω.2.1.2 An alternative definition of viscosity solutionThe definition of viscosity solutions based on test functions is probably the mostpopular and useful. However, the original definition was based the notion on subandsuperdifferentials which are standard tools in convex analysis.To introduce this notion, we first recall the elementary definition of a differentiablefunction:Definition 2.7. v is differentiable at x 0 if and only if there exists a linear operatorA such thatv(y) − v(x 0 ) − A(y − x 0 )lim= 0y→x 0 |y − x 0 |

✐✐24 Chapter 2. Viscosity Solutions of First Order PDEsIf v is not differentiable at x 0 we can still have a generalized version of thedifferential, based on the following idea (sketched in one dimension for simplicity).In a univariate differentiable function the tangent at x 0 is uniquely defined bytaking the limit of secants passing through points of the graph of v which convergeto x 0 , the limit slope defining the derivative v ′ . If v is not differentiable at x 0 , thisprocedure does not select a unique tangent (and a unique derivative), but rathera set of “generalized tangents”. For example, v(x) = |x| has a unique tangent forevery x ≠ 0, whereas for x = 0 the “generalized tangents” are all the straight linespassing in (0, 0) and not crossing the graph of v, i.e. the lines of the set{y = mx, m ∈ (−1, 1)}.The standard extension of derivatives used in nonsmooth analysis is based on thisidea and corresponds to the following definition of sub- and superdifferentials.Definition 2.8. Let Ω be an open set of R d and v : Ω → R.The super-differential D + v(x) of v at x ∈ Ω, is defined as the set:{}D + v(x) := p ∈ R d v(y) − v(x) − p · (y − x): lim sup≤ 0y→x y∈Ω |y − x|The sub-differential D − v(x) of v at x ∈ Ω, is defined as the set:{}D − v(x) := q ∈ R d v(y) − v(x) − q · (y − x): lim inf≥ 0 .y→x y∈Ω |y − x|We remark that the sub- and superdifferential are (possibly empty) convexand closed subsets of R d . Moreover, if at the point x we haveD + v(x) = D − v(x)then v is differentiable in the sense of Definition 2.7. Going back to the example ofv(x) = |x|, we have, for x ≠ 0:{D + v(x) = D − −1 for x < 0v(x) =1 for x > 0so they coincide and give the standard derivative.At x = 0, we have{D − v(0) = q ∈ R d : lim infTaking for example q = 1, we have :=y→0|y| − q y|y|}≥ 0 ={}q ∈ R d : lim inf (1 − q sgn(y)) ≥ 0 .y→0lim inf(1 − sgn(y)) = min{2, 0} = 0.y→0In the same way, we can show that every q ∈ [−1, 1] belongs to the subdifferential.Then, D − v(0) = [−1, 1].

✐✐2.1. The definition of viscosity solution 25Considering now the superdifferential at x = 0, we have:{}D + v(0) = p ∈ R d |y| − p y: lim sup ≤ 0 =y→0 |y|{}= p ∈ R d : lim sup(1 − p sgn(y)) ≤ 0y→0On the other hand, we havelim sup(1 − p sgn(y)) = lim(1 + max(p, −p)) =y→0y→0= lim(1 + |p|) ≥ 1.y→0This shows that D + v(0) = ∅.We can now give the definition of viscosity solution based on sub- and superdifferentials.Definition 2.9. Let u ∈ BUC(Ω). We say that u is a viscosity solution of (2.1)if and only if, for any ϕ ∈ C 1 (Ω), the following conditions hold:(i) H(x, u(x), p) ≤ 0, for all p ∈ D + u(x), x ∈ Ω(i.e., u is a viscosity subsolution);(ii) H(x, u(x), p) ≥ 0, for all p ∈ D − u(x), x ∈ Ω(i.e., u is a viscosity supersolution).The following result explains the link between the two definitions.Theorem 2.10. Let Ω be an open subset of R d and u ∈ C(Ω), then(i) p ∈ D + u(x), x ∈ Ω if and only if there exists a function φ ∈ C 1 (Ω) such thatDφ(x) = p and u − φ has a local maximum point at x.(ii) q ∈ D − u(x), x ∈ Ω if and only if there exists a function φ ∈ C 1 (Ω) such thatDφ(x) = q and u − φ has a local minimum point at x.Proof. A proof of this theorem can be found in [BCD97] (Lemma 1.7, p. 29).We conclude this section with a further result linking viscosity and a.e. solutions.Theorem 2.11. The following statements hold true:(i) if u ∈ C(Ω) is a viscosity solution of (2.1), thenH(x, u(x), Du(x)) = 0at any point x ∈ Ω where u is differentiable;(ii) if u is locally Lipschitz continuous and is a viscosity solution of (2.1), thenH(x, u(x), Du(x)) = 0a.e. in Ω

✐✐26 Chapter 2. Viscosity Solutions of First Order PDEsProof. Let u be a point of differentiability for u. Then, D + u(x) ∩ D − u(x) ≠ ∅since this set contains Du(x). Moreover, this intersection reduces to a singleton, sothatDu(x) = D + u(x) = D − u(x).Hence, by Definition 2.9 we have0 ≥ H(x, u(x), Du(x)) ≥ 0which proves (i). Statement (ii) follows immediately from (i) and Rademacher’stheorem (which states that Lipschitz continuous functions are a.e. differentiable,see [E98] pp. 280–281 for the proof).2.1.3 Uniqueness of viscosity solutionsThe crucial point in the theory of viscosity solution (and also the essential advantageover the concept of solutions almost everywhere) is to prove uniqueness. This isdone via a comparison principle, also termed maximum principle.Theorem 2.12. Let u, v ∈ BUC(Ω) be respectively a sub- and a supersolution for(2.1), and letu(x) ≤ v(x) for any x ∈ ∂Ω.Then,u(x) ≤ v(x) for any x ∈ Ω.This is enough to get uniqueness, since taken two viscosity solutions of theequation they are (both) sub- and supersolutions. This implies that both u(x) ≤v(x) and u(x) ≥ v(x) for any x ∈ Ω, i.e., u(x) = v(x) in Ω.The following is a classical assumption for uniqueness.(A4) Let ω(·) be a modulus of continuity. We assume that|H(x, u, p) − H(y, u, p)| ≤ ω(|x − y|(1 + |p|))Q R (x, y, u, p)for any x, y ∈ Ω, u ∈ [−R, R] and p ∈ R d , whereQ R (x, y, u, p) ≡ max (ϕ(H(x, u, p)) , ϕ (H(y, u, p)))and ϕ : R → R + is continuous.We can now give a sufficient condition for the comparison principle to hold.Theorem 2.13. Let assumptions (A1)–(A4) be satisfied. Then, the comparisonprinciple (Theorem 2.12) holds for (2.1), i.e., the viscosity solution is unique.Proof. We only sketch the proof, which is carried out via variable doubling.The goal is to prove that M = max x∈Ω(u−v) is negative. To this end, we introducea test function depending on two variables:ψ ε (x, y) ≡ u(x) − v(y) −|x − y|2ε 2 .

✐✐2.2. Viscosity solution for evolutive equations 27Due to the penalization term, we can expect that the maximum points (x ε , y ε ) forψ ε should have x ε and y ε close enough for ε small. Moreover, for ε → 0 + we have:M ε → M,|x − y| 2ε 2 → 0,u(x ε ) − v(y ε ) → M.This allows to pass to the limit and get the comparison result.2.2 Viscosity solution for evolutive equationsOnce fixed some basic ideas, the definition of viscosity solution can be easily extendedto the evolutive case, by taking into account also the time derivative.Definition 2.14. u ∈ BUC(Ω × (0, T ) is a viscosity solution in Ω × (0, T ) of theequationu t + H(x, t, u, Du) = 0 (2.18)if and only if, for any ϕ ∈ C 1 (Ω × (0, T )), the following conditions hold:(i) at every point (x 0 , t 0 ) ∈ Ω × (0, T ), local maximum for u − ϕ,ϕ t (x 0 , t 0 ) + (H(x 0 , t 0 , u(x 0 , t 0 ), Dϕ(x 0 , t 0 )) ≤ 0(i.e., u is a viscosity subsolution).(ii) at every point (x 0 , t 0 ) ∈ Ω × (0, T ), local minimum for u − ϕ,ϕ t (x 0 , t 0 ) + (H(x 0 , t 0 , u(x 0 , t 0 ), Dϕ(x 0 , t 0 )) ≥ 0(i.e., u is a viscosity supersolution).As one can see, this definition is very close to the one used for stationaryproblems. In fact, it could also be derived from Definition 2.1 by setting the problem(2.1) in d + 1 variables (x, t) ∈ R d × (0, T ) and making the time derivative explicit.The following comparison result refers to the evolutive problem and usuallygives the uniqueness for the particular case H(x, t, u, Du) = H(t, Du).Theorem 2.15. Assume H ∈ C((0, T ) × R d ). Let u 1 , u 2 ∈ C(R d × (0, T )) be,respectively, viscosity sub- and supersolution in R d × [0, T ] of the equationu t (x, t) + H(t, Du(x, t)) = 0.Then,sup (u 1 − u 2 ) ≤ sup(u 1 (·, 0) − u 2 (·, 0)).R d ×(0,T )R dProof. The proof can be found in [BCD97], p. 56. Note that more generalcomparison results for more general Hamiltonians can be found in [Ba98].

✐✐28 Chapter 2. Viscosity Solutions of First Order PDEs2.2.1 Representation formulae and Legendre transformIn some cases it is possible to derive representation formulaa for the viscosity solution.This formulae have a great importance from both the analytical and thenumerical point of view, and will be derived here in two major cases: linear advectionequations and convex HJ equations.Linear advection equationLet us start by the linear case in which the representation formula can be obtainedvia the method of characteristics.Theorem 2.16. Let u : R d × (t 0 , T ) → R be a viscosity solution of the initial valueproblem{u t (x, t) + λu(x, t) + f(x, t) · Du(x, t) = g(x, t) (x, t) ∈ R d × (t 0 , T )u(x, t 0 ) = u 0 (x) x ∈ R d (2.19).Assume that λ ≥ 0, f : R d × (t 0 , T ) → R d and g : R d × (t 0 , T ) → R are continuousin (x, t) and f is globally Lipschitz continuous with respect to x. Then,∫ tu(x, t) = e λ(t0−t) u 0 (y(x, t; t 0 )) + e λ(s−t) g(y(x, t; s), s)dst 0(2.20)where y(x, t; s) is the the position at time s of the solution trajectory passing throughx at time t, i.e., solving the Cauchy problem⎧⎨ dy(x, t; s) = f(y(x, t; s), s)ds (2.21)⎩y(x, t; t) = x.Proof. We give the proof under the additional assumption that u ∈ C 1 .Let (x, t) be fixed, and denote for shortness the solution of (2.21) as y(s) ≡ y(x, t; s).Writing the equation in (2.19) at a point (y(s), s) and multiply by e λs , we havee λs u s (y(s), s) + λe λs u(y(s), s) + e λs f(y(s), s) · Du(y(s), s) = e λs g(y(s), s).Since u is differentiable, this may also be rewritten asIntegrating (2.22) over the interval [t 0 , t] we getd [e λs u(y(s), s) ] = e λs g(y(s), s). (2.22)ds∫ te λt u(y(t), t) = e λt0 u(y(t 0 ), t 0 ) + e λs g(y(s), s)ds.t 0(2.23)Recalling that y(t) = y(x, t; t) = x and u(y(t 0 ), t 0 ) = u 0 (y(t 0 )) and dividing by e λt ,we get∫ tu(x, t) = e λ(t0−t) u 0 (y(t 0 )) + e λ(s−t) g(y(s), s)dst 0

✐✐2.2. Viscosity solution for evolutive equations 29which coincides with (2.20).We recall that a solution of (2.21) is called characteristic curve. Note that, inusing (2.20), (2.21) is integrated backwards. Note also that, if λ is strictly positive,f and g do not depend on t and the source g is bounded, then u(x, t) has a limitfor t − t 0 → ∞. Setting conventionally t = 0 and letting t 0 → −∞, we obtain infact the limitu(x) ==∫ 0−∞∫ ∞0e λs g(y(x, 0; s))ds =e −λs g(y(x, 0; −s))ds (2.24)which is a regime solution for problem (2.19), or, in other terms, solves the stationaryequationλu(x) + f(x) · Du(x) = g(x)for x ∈ R d .Hamilton–Jacobi equationsConcerning HJ equations, the representation formula is known as Hopf–Lax formula,and is typically related to the problem:{u t + H(Du) = 0 (x, t) ∈ R d × (0, T ),u(x, 0) = u 0 (x) x ∈ R d (2.25),where H : R d → R is convex and satisfies the coercivity condition:H(p)lim = +∞. (2.26)|p|→+∞ |p|Assumption (2.26) allows to give the following definition:Definition 2.17. Let (2.26) be satisfied. We define the Legendre–Fenchel conjugate(or Legendre–Fenchel transform) of H for q ∈ R d as:H ∗ (q) = supp∈R d {p · q − H(p)}. (2.27)Note that the convexity assumption on H implies that H is continuous, andbeing also coercive in the sense of (2.26), the sup in (2.27) is in fact a maximum.In general, Legendre–Fenchel transform may not allow for an explicit computation– we will show in Chapter 3 a numerical procedure to approximate it. Afew examples, anyway, can be computed analytically, among which the quadraticHamiltonian:H 2 (p) = |p|22 ,for which an easy computation givesH ∗ 2 (q) = |q|22 .

✐✐30 Chapter 2. Viscosity Solutions of First Order PDEsIt is interesting to note that a similar construction may also work for the noncoercivecase, but in this case the Legendre–Fenchel conjugate will not in general be boundedand defined everywhere. For example, takingH 1 (p) = |p|and using the definition (2.27), it is easy to check thatH ∗ 1 (q) ={0 for |q| ≤ 1+∞ elsewhere.The main result of interest here concerns two important properties of the Legendretransform:Theorem 2.18. Let (2.26) be satisfied. Then, the function H ∗ has the followingproperties:(i) H ∗ : R d → R is convex and(ii) H(p) = H ∗∗ (p), for any p ∈ R dH(p)lim = +∞;|p|→+∞ |p|Proof. The proof can be found in [E98], p. 122.The above theorem says that the by applying twice the Legendre transformto H we obtain back H itself. The definition of Legendre transform is very usefulto characterize the unique solution of (2.25) by means of the so-called Hopf–Laxrepresentation formula. The following theorems provide the basic results concerningthis characterization.Theorem 2.19. The function u defined by the following Hopf–Lax formula:[( )] x − yu(x, t) = inf v 0 (y) + tH ∗y∈R d t(2.28)is Lipschitz continuous, differentiable almost everywhere in R d ×(0, +∞) and solvesa.e. the initial value problem (2.25).Proof. The proof relies on the functional identity{ ( ) }x − yu(x, t) = min (t − s)H ∗ + u(y, s)y∈R d t − s(2.29)(which is proved in [E98] p. 126). This identity implies that u is Lipschitz continuousand, by Rademacher’s theorem, differentiable almost everywhere. Now, consider apoint of differentiability (x, t).

✐✐2.2. Viscosity solution for evolutive equations 31Step 1. u is a subsolution at (x, t).Fix p ∈ R d and and a positive parameter h. By (2.29), we have{ ( ) }x + hp − yu(x + hp, t + h) = min hH ∗ + u(y, t) ≤ hH ∗ (p) + u(x, t)y∈R d hwhich impliesNow, for h → 0, we getu(x + hp, t + h) − u(x, t)h≤ H ∗ (p)p · Du(x, t) + u t (x, t) ≤ H ∗ (p).This inequality holds for any p ∈ R d , so that we haveu t (x, t) + H(Du(x, t)) + u t (x, t) + maxp∈R d {p · Du(x, t) − H ∗ (p)} ≤ 0where we have used the fact that H = H ∗∗ .Step 2. u is a supersolution at (x, t).Let us choose z such thatu(x, t) = t H ∗ ( x − zt)+ u 0 (z).Fix a positive parameter h and set s = t − h, y = x s/t + (1 − s/t)z. This impliesand thereforeu(x, t) − u(y, s) ≥ tH ∗ ( x − ztAs a consequence,and, for h → 0, we obtain:Then, we finally getx − st)= (t − s)H ∗ ( x − zt= y − z ,s[+ u 0 (z) −).sH ∗ ( y − zsu(x, t) − u (( )1 − h t x +h( )tz, t − h) x − z≥ H ∗ ,htx − zt( ) x − z· Du(x, t) + u t (x, t) ≥ H ∗ .t) ]+ u 0 (z) =u t (x, t) + H((Du(x, t)) = u t (x, t) + max {p · Du(x, t) − H ∗ (p)} ≥p∈R d≥ u t (x, t) + x − z( ) x − z· Du(x, t) − H ∗ ≥ 0tt

✐✐32 Chapter 2. Viscosity Solutions of First Order PDEswhich concludes the proof.Theorem 2.20. The unique viscosity solution of (2.25) is given by the Hopf-Laxrepresentation formula.Proof. The proof of this result can be found in [E98] p. 561.We finally note that an equivalent way of rewriting Hopf-Lax formula is to seta := x − ytso that y = x + at. Since the minimization with respect to y corresponds to aminimization with respect to a, we also obtain:u(x, t) = infa∈R d [u 0 (x − at) + tH ∗ (a)] . (2.30)This formula shows more clearly the link between the value of the solution at (x, t)and the values of the solution at points (x−at, 0) and it will be useful in the analysisof semi-Lagrangian schemes.2.2.2 Semiconcavity and regularity of viscosity solutionsAs we have seen earlier in this chapter, we expect viscosity solutions to be in generalonly uniformly continuous. However, some further regularity results can be given.We will examine here the most typical of these results. A regularity property whichplays an important role in HJ equations is semiconcavity, defined as follows.Definition 2.21. A function u ∈ C(Ω) is semiconcave in the open convex set Ω ifthere exists a constant C > 0 such that, for any x, z ∈ Ω and µ ∈ [0, 1]:µu(x) + (1 − µ)u(y) ≤ u(µx + (1 − µ)y) + 1 2 Cµ(1 − µ)|x − y|2 (2.31)This definition corresponds to the request that u(x) − 1 2 C|x|2 is concave, as itcan be immediately verified. Note that an equivalent definition, when u is continuous,is to require thatu(x + h) − 2u(x) + u(x − h) ≤ C|h| 2 (2.32)for all x ∈ Ω and h ∈ R d , for |h| sufficiently small.Clearly, concave functions are also semiconcave. Moreover, C 1 functions havinga Lipschitz continuous gradient are semiconcave. Another important exampleof nondifferentiable semiconcave functions is given by marginal functions, which arefunctions defined asu(x) = inf F (x, a) (2.33)a∈Aprovided F (·, a) satisfies (2.31) uniformly with respect to a. Marginal functionsare particularly important in control problems, since value functions belong to thisclass. We will see that semiconcavity is one of the properties that the value functioninherits from the data in many control problems.

✐✐2.2. Viscosity solution for evolutive equations 33Example: distance function from a setas usual the distance d(x, C) byd(x, C) := inf |x − y|y∈CGiven a domain C ⊂ R d , C ≠ ∅, defineThen d 2 is semiconcave in R d since the application x → |x − y| 2 is C ∞ and hasconstant second derivatives. Moreover, d itself is semiconcave on any compact sethaving strictly positive distance from C, because the mapping x → |x − y| hasalso bounded second derivatives in such a set, and that bound is uniform for y ina bounded set. It is interesting to note that our counterexample 2.2 correspondsto the characterization of the distance function from the set R \ (−1, 1) and onlyviscosity solution u(x) = 1 − |x| is semiconcave (and is also the only semiconcavea.e. solution).Properties of semiconcave functions We review here some properties of semiconcavefunctions which are of particular interest in relationship with viscosity solutionsof HJ equations.Theorem 2.22. Let u be semiconcave in Ω. Then, u is locally Lipschitz continuousin Ω.Proof. For any fixed x ∈ Ω and any h such that x + h ∈ Ω, we haveu(x + h) − u(x) = ψ(x + h) − ψ(x) + Cx · h + C 2 |h|2where ψ(x) = u(x) − 1 2 C|x|2 is concave and, therefore, Lipschitz continuous in Ω.Since the sum of the last two terms is quadratic in h, this proves the assertion.Theorem 2.23. Let us assume that the Hamiltonian is convex and that the initialcondition u 0 is semiconcave. Then, the viscosity solution of (2.25) si semiconcave.Proof. Assume that u 0 satisfies (2.32), and let x and h be fixed. By the Hopf–Laxformula, the solution u(x, t) of (2.25) may be written at x and x ± h as:u(x, t) = inf [u 0 (x − at) + tH ∗ (a)] = u 0 (x − at) + tH ∗ (a)a∈R du(x + h, t) = inf [u 0 (x + h − at) + tH ∗ (a)] = u 0 (x + h − a + t) + tH ∗ (a + )a∈R du(x − h, t) = inf [u 0 (x − h − at) + tH ∗ (a)] = u 0 (x − h − a − t) + tH ∗ (a − ),a∈R dwhere we have denoted by a, a + and a − the minimizers for respectively x, x−h andx + h in the Hopf–Lax formula. On the other hand, replacing a ± by a, we obtainthe inequalitiesso that at last we get:u(x + h, t) ≤ u 0 (x + h − at) + tH ∗ (a)u(x − h, t) ≤ u 0 (x − h − at) + tH ∗ (a),u(x + h) − 2u(x) + u(x − h) ≤ u 0 (x + h − at) − 2u 0 (x − at) + u 0 (x − h − at) ≤≤ C|h| 2

✐✐34 Chapter 2. Viscosity Solutions of First Order PDEsand, by semiconcavity of u 0 , we obtain semiconcavity of uLet now u ∈ W 1,∞ (Ω), and define the setlocD ∗ u(x) := {p ∈ R d : p = limn→∞ Du(x n), x n → x}.Then, D ∗ is nonempty and closed for any x ∈ Ω. Denote by co D ∗ u(x) its convexhull. A classical result in nonsmooth analysis (see [Cl83] for the proof) states thatco D ∗ u(x) = ∂u(x),∀x ∈ Ωwhere ∂u(x) is the generalized gradient of u at x, defined by∂u(x) := {p ∈ R d : u(x, q) ≥ p · q, ∀q ∈ R d } = (2.34)= {p ∈ R d : u(x, q) ≤ p · q, ∀q ∈ R d }and u, u are respectively, the generalized directional derivatives defined byu(y + tq) − u(y)u(x; q) := lim supy→x,t→0 t+u(x; q) :=u(y + tq) − u(y)lim infy→x,t→0 + t(2.35)An important property is that, under the assumption of semiconcavity, some generalizedand classical derivatives coincide.Proposition 2.24. Let u be semiconcave in Ω. Then, for all x ∈ Ω:(i) D + (u(x) = ∂u(x) = co D ∗ u(x);(ii) Either D − u(x) = ∅ or u is differentiable at x;(iii) If D + u(x) is a singleton, then u is differentiable at x;(iv) ∂u∂q (x) = min p∈D + u(x) p · q for all unit vectors q.Proof. The proof can be found in [BCD97] p. 66.Remark 2.25. Claim (ii) has an important consequence. Assume that the Legendretransform H ∗ is coercive and has bounded second derivatives. Then, writing theHopf–Lax representation formula for u,u(x, t) = infa∈R d [u 0 (x − at) + tH ∗ (a)] ,we note that the infimum is actually a minimum, and that the function in squarebrackets to be minimized is semiconcave. At a minimum point a, its subdifferentialcannot be empty (in fact, it must contain at least the origin), and by the regularityof H ∗ , claim (ii) implies that u must be differentiable at the point x − at.Moreover, semiconcavity allows to clarify the link between a.e. solutions andviscosity supersolutions.

✐✐2.3. Problems in bounded domains 35Proposition 2.26. Let u be semiconcave and such thatH(x, u(x), Du(x)) ≥ 0 a.e. in Ω,where H is continuous. Then, u is a viscosity supersolution ofH(x, u(x), Du(x)) ≥ 0 in Ω,Proof. The proof can be found in [BCD97].Note that this proposition gives a sharper interpretation of counterexample(2.2). In fact, all a.e. solutions are subsolutions in the viscosity sense, but only thesemiconcave one u(x) = 1 − |x| is also a supersolution.Under suitable assumptions, it is possible to prove the semiconcavity of viscositysolutions for HJ equations of more general structure, as the following theoremshows.Theorem 2.27. Let λ > 0 and u ∈ W 1,∞ be a viscosity solution ofλu(x) + H(x, Du(x)) = 0with Lipschitz constant L u . Assume that H satisfiesx ∈ R d|H(x, p) − H(x, q)| ≤ ω(|p − q|), ∀x, p, q ∈ R d (2.36)and that, for some C > 0 and ̂L > 2L u ,H(x + h, p + Ch) − 2H(x, p) + H(x − h, p − Ch) ≥ −C|h| 2 (2.37)holds for all x, hR d , p ∈ B(0, ̂L). Then, u is semiconcave on R d .Proof. The proof can be found in [BCD97] p. 69.A final remark is that combining convexity of the Hamiltonian with semiconcaveregularity one can obtain an unexpected differentiability result for the viscositysolution.Proposition 2.28. Assume that u ∈ C(Ω) is a viscosity solution in Ω ofλu(x) + H(x, Du(x)) = 0with λ ≥ 0. Assume also that H(x, ·) is strictly convex for any fixed x ∈ Ω and that−u is semiconcave. Then u ∈ C 1 (Ω).Proof. The proof can be found in [BCD97].2.3 Problems in bounded domainsAnother peculiar point of the theory of viscosity solutions is the way boundaryconditions are satisfied. It is not surprising that boundary conditions cannot bearbitrarily assigned in first-order equations – even for advection equations we knowthat Dirichlet data can only be imposed at points of the boundary where the drift isdirected inwards. The technical point for nonlinear equations is that the equationplays a role up to the boundary.

✐✐36 Chapter 2. Viscosity Solutions of First Order PDEs2.3.1 Boundary Conditions in a weak senseIn the theory of viscosity solutions, the typical compact form for boundary conditionsismin(H(x, u(x), Du(x)), B(x, u(x), Du(x))) ≤ 0max(H(x, u(x), Du(x)), B(x, u(x), Du(x))) ≥ 0where x ∈ ∂Ω, and B represents a suitably defined boundary operator. In order toexplain this setting, we will consider in detail the main situations of interest. Forexample, the Dirichlet condition would be formulated in a classical form as:u(x) = b(x)(x ∈ ∂Ω).In its weak formulation, this corresponds to a boundary operator defined byB(x, u(x), Du(x))) ≡ u − b.We still remark that not all the boundary conditions are compatible with the equation.In this section, we briefly analyze the effect of Dirichlet, Neumann and “stateconstraints” boundary conditions, posed on parts of the boundary. First, note thatboundary conditions should be imposed in a weak sense. The condition whichdefines u as a viscosity subsolution for (2.1) requires that for any test functionϕ ∈ C 1 (Ω) and x ∈ ∂Ω local maximum point for u − ϕ,min{H(x, u(x), Dϕ(x)), B(x, u, Dϕ(x))} ≤ 0 (2.38)for a given boundary operator B. Similarly, the condition for supersolutions requiresthat for any test function ϕ ∈ C 1 (Ω) and x ∈ ∂Ω local minimum point for u − ϕ,max{H(x, u(x), Dϕ(x)), B(x, u, Dϕ(x))} ≥ 0. (2.39)The effect of the Dirichlet condition is to impose a value on u according to theabove conditions, in particular the value u(x) = b(x) is set at every point whereH(x, u(x), Dϕ(x)) ≥ 0 (for subsolutions) and H(x, u(x), Dϕ(x)) ≤ 0 (for supersolutions).The Neumann conditon, which is classically stated as∂u(x) = m(x)∂ν x∈ ∂Ωuses in its weak formulation the boundary operatorB(x, u(x), Du(x))) ≡ ∂u∂ν − m.where ν(·) represents the outward normal to the domain Ω. A typical use of thiscondition occurs when we know (or presume) that the level curves of the surface areorthogonal to the boundary ∂Ω or to a part of it, in which case we simply choosem(x) = 0.On the other hand, in the “state constraints” boundary condition neither avalue for u nor a value for its normal derivative ∂u/∂ν(x) is imposed. In this respect,it has been sometimes interpreted as a “no boundary condition” choice althoughthis interpretation is quite sloppy. In fact, a bounded and uniformly continuous

✐✐2.3. Problems in bounded domains 37function u is said to be a state constrained viscosity solution if and only if it is asubsolution in Ω and a supersolution in Ω (i.e., up to the boundary). Whenever abound on the L ∞ norm of the solution is known, this condition can be also stated asa Dirichlet boundary condition by simply setting b(x) ≡ C, provided the constantC satisfiesC > maxx∈Ω u(x) .By this choice (2.38) is trivially satisfied, whereas (2.39) requires (strictly)Representation formulae in bounded domainsin a bounded domain Ω. Define byH(x, u(x), Dϕ(x)) ≥ 0. (2.40)We treat separately the linear caseΓ in := {x ∈ ∂Ω : f(x, t) · ν(x) < 0} (2.41)the subset of boundary points where the vectorfield f is pointing inward Ω. Weconsider again the linear evolutive problem (1.1), which is rewritten here:{u t + f(x, t) · Du = g(x, t) (x, t) ∈ Ω × (t 0 , T )u(x, 0) = u 0 (x)x ∈ Ωand complemented with the Dirichlet boundary conditionu(x, t) = b(x, t) x ∈ Γ in . (2.42)Note that, in terms of characteristics, the existence of an inflow boundary Γ inmeans that the characteristic y(x, t; s) may have a finite exit time from Ω for sgoing backwards. More precisely, we will set:θ(x, t) := sup { s ≤ t : y(x, t; s) ∈ R N \ Ω } . (2.43)By its definition, θ(x, t) represents the time at which the characteristic passing at(x, t) starts from the boundary. The representation formula (written at (x, t)) maybe changed accordingly, obtainingu(x, t) =∫ tt 0∨θ(x,t)g(y(x, t; s), s)ds + u(y(x, t; t 0 ∨ θ(x, t)), t 0 ∨ θ(x, t)), (2.44)where a ∨ b = max(a, b), finite or −∞. Despite its formal complexity, (2.44) hasa natural interpretation. If θ(x, t) < t 0 , that is, if the characteristic has run inthe interior of Ω during the time interval (t 0 , t), the solution u(x, t) has the usualrepresentation, i.e.u(x, t) =∫ tt 0g(y(x, t; s), s)ds + u(y(x, t; t 0 ), t 0 ).On the other hand, if θ(x, t) ≥ t 0 , it happens that the characteristic passing at (x, t)has started from the boundary, so the value which is propagated is the boundarycondition, computed at the intersection between the characteristic and the boundaryitself:u(x, t) =∫ tθ(x,t)g(y(x, t; s), s)ds + b(y(x, t; θ(x, t)), θ(x, t)).This generalized version of the representation formula can be useful in assigning aDirichlet boundary condition, and in fact will be used in the numerical implementationof SL schemes.

✐✐38 Chapter 2. Viscosity Solutions of First Order PDEs2.4 Viscosity solutions and entropy solutionsIt is interesting to remark that there exists a link between viscosity solutions and entropysolutions which are the usual analytical tool in the framework of conservationlaws. Although this link is valid only in one space dimension and for a particularclass of Hamiltonians, it is the basis for some interesting remarks and also hasan important role in the construction of numerical schemes for Hamilton–Jacobiequations.Consider the two problems, an evolutive Hamilton–Jacobi equation{v t + H(v x ) = 0 (x, t) ∈ R × (0, T ),(2.45)v(x, 0) = v 0 (x) x ∈ R(where the Hamiltonian H is assumed to be convex) and the associated conservationlaw {u t + H(u) x = 0 (x, t) ∈ R × (0, T ),(2.46)u(x, 0) = u 0 (x) x ∈ R.Assume now thatv 0 (x) =∫ x−∞u 0 (ξ)dξ.Then, it turns out that this relationship between u and v is preserved also for t > 0,that is, if u is the entropy solution of (2.46), thenv(x, t) =∫ x−∞u(ξ, t)dξis the unique viscosity solution of (2.45). Viceversa, if v is the viscosity solution of(2.45), then u = v x is the unique entropy solution for (2.46).Note that v is a.e. differentiable, and the singular points for its derivativev x correspond to shocks for u. Note also that semiconcavity of v correspond to aone-sided (positive) bound on the derivative of u. This link will be also useful fornumerical purposes.We briefly review the main results.Proposition 2.29. Let H ∈ C(R) be convex, and assume that v ∈ W 1,∞ (R×(0, T )is a solution of (2.45). Then, u := v x is a weak solution of (2.46).Proof. The proof can be found in [Li82], p.268.Theorem 2.30. Let H ∈ C 1 (R) be convex, v 0 ∈ W 1,∞ (R). If v ∈ W 1,∞ (R×(0, T ))is the unique viscosity solution of (2.45), then u := v x is the unique entropy solutionof (2.46).Proof. As we know, the viscosity solution v is the limit in L ∞ (R × (0, T )), asε → 0 + , of regular solutions v ε of the following problem{vt ε (x, t) + H(vx) ε = εvxx ε (x, t) ∈ R × (0, T )v ε (x, 0) = v 0 (x) x ∈ R,

✐✐2.4. Viscosity solutions and entropy solutions 39Hence we have, for any ϕ ∈ C ∞ (R × (0, T )),∫ T ∫∫ T ∫lim vx(x, ε t)ϕ(x, t) dx dt = − lim v ε (x, t)ϕ x (x, t) dx dt =ε→00 Rε→00 R∫ T ∫∫ T ∫= − lim v(x, t)ϕ(x, t) dx dt = v x (x, t)ϕ(x, t) dx dt.ε→00 R0 RObviously, the function u ε := vx ε solves the derived problem{u ε t (x, t) + H(u ε ) x = εu ε xx (x, t) ∈ R × (0, T )u ε (x, 0) = v 0x (x) x ∈ R.Since the sequence u ε converges in L 1 loc (R × (0, T )) for ε → 0+ , it must converge tothe entropy solution u of (2.46). This implies that, for any ϕ ∈ C ∞ (R × (0, T )),∫ T ∫∫ T ∫u ε (x, t)ϕ(x, t) dx dt = u(x, t)ϕ(x, t) dx dt.limε→0As a consequence,∫∫ T0and v x = u a.e. in R × (0, T )).0RRv x (x, t)ϕ(x, t) dx dt =∫ T00∫RRu(x, t)ϕ(x, t) dx dt,A converse of this result is also true. The first step is an intermediate result:Proposition 2.31. Let H ∈ C(R) be convex, and assume that u ∈ L ∞ loc(R × (0, T ))is a weak solution of (2.46). Definev(x, t) :=∫ xαu(ξ, t) dξ (2.47)for a fixed α ∈ R. Then, v ∈ W 1,∞loc(R × (0, T )) and v is a solution of (2.45) almosteverywhere.Proof. Since u ∈ L ∞ loc(R × (0, T )), there exists a set A ⊆ (0, T ) of zero Lebesguemeasure, such that for any t ∈ (0, T )\A, u is defined a.e. on R and u(·, t) ∈ L ∞ loc (R).Then, for such values of t, v(·, t) ∈ L ∞ loc(R). Morevover, for any t ∈ (0, T ) \ A andany ϕ ∈ C0 ∞ (R × (0, T )),∫v(x, t)ϕ x (x, t) dx =∫ [∫ xRR∫α= − u(x, t)ϕ(x, t) dt.R]u(ξ, t) dξ ϕ x (x, t) dξ =Thus, integrating on (0, T ), one has u = v x in the sense of distributions and a.e.Since u is a weak solution of (2.46), we have, for any ϕ ∈ C0 ∞ (R × (0, T )),∫ T ∫∫ T ∫H(u) ϕ x (x, t) dx dt = − u(x, t)ϕ t (x, t) dx dt =0= −R∫ T0∫Rv x (x, t)ϕ t (x, t) dx dt =0 R∫ T0∫Rv(x, t)ϕ tx (x, t) dx dt

✐✐40 Chapter 2. Viscosity Solutions of First Order PDEsSo there exists v t in the sense of distributions, and v t = −H(u) = −H(v x ). Therefore,v ∈ W 1,∞loc(R × (0, T )), and v is a solution almost everywhere of (2.45).Now, for any u ∈ C([0, T ]; L 1 (R)) we definev(x, t) :=∫ x−∞u(ξ, t) dξ . (2.48)Clearly, for any t ∈ (0, T ), the function v(·, t) is absolutely continuous and v x = ua.e.. We can now prove the main result.Theorem 2.32. Let H ∈ C 1 (R) and u 0 ∈ L ∞ (R) ∩ L 1 (R). Assume that u ∈L ∞ (R × (0, T )) ∩ C([0, T ]; L 1 (R)) is the unique entropy solution of (2.46). Then,the function v given by (2.48) is the unique viscosity solution of (2.45) for the initialconditionv 0 (x) :=∫ x−∞u 0 (ξ) dξ.Proof. Since u ∈ C([0, T ]; L 1 (R)), we have that v ∈ L ∞ (R × [0, T ]). As in theprevious proposition, it is easy to show that v ∈ W 1,∞ (R × (0, T )) and that v is asolution a.e. of (2.45). Moreover,∫ xlim |v(x, t) − v 0(x)| ≤ lim |u(ξ, t) − u 0 (ξ)| dξ = 0.t→0 t→0−∞Now, suppose that v is not the viscosity solution of (2.45), and denote by v theunique viscosity solution. Then, by Theorem 2.30, v x is the unique entropy solutionof (2.46) and therefore, for any ϕ ∈ C ∞ 0 (R × (0, T )),∫ ∫(v − v)ϕ x dx dt = 0.Since ϕ can be arbitrarily chosen, the conclusion follows.2.5 Discontinuous viscosity solutionsWe conclude this chapter giving some ideas about the extension of the “classical”theory of viscosity solutions to the discontinuous case. This extension has beenstrongly motivated by a number of applications, e.g. to image processing and togames, where it is natural to accept discontinuous solutions.Let w : R d → R be a bounded function. We recall two definitions which willplay a key role in defining a bounded, but possibly discontinuous viscosity solution.The lower semi-continuous envelope of w is defined asw(x) := lim inf w(y),y→xwhereas the upper semi-continuous envelope asw(x) := lim sup w(y).y→x

✐✐2.6. Commented references 41With this additional tool, we can extend the notion of viscosity solutions as follows.Definition 2.33. A function u ∈ L ∞ (Ω) is a viscosity solution in Ω ofH(x, u, Du) = 0if and only if, for any ϕ ∈ C 1 (Ω) the following conditions hold:(i) at every point x 0 ∈ Ω, local maximum for u − ϕ,H(x 0 , u(x 0 ), Dϕ(x 0 )) ≤ 0(i.e., u is a viscosity subsolution);(ii) at every point x 0 ∈ Ω, local minimum for u − ϕ,i.e., u is a viscosity supersolution.H(x 0 , u(x 0 ), Dϕ(x 0 )) ≥ 0It is important to note that the extension of the comparison principle to thisnew settings is not trivial. However, in many relevant problem one can get a uniquenessresult for lower semicontinuous viscosity solution. A comprehensive introductionto these results goes beyond the scopes of this book and can be found in [Ba93].2.6 Commented referencesThe theory of viscosity solutions has started with the papers by Crandall and Lions[CL84] on the solution of first order Hamilton–Jacobi equations. However, a veryimportant source of inspiration for this theory had come from some earlier papers byKružkov [Kr66], in which the solution for first order problems was obtained via thevanishing viscosity methods (i.e., by elliptic/parabolic regularization). These paperswere strongly related to the analysis of conservation laws, where Kružkov gaveimportant contributions (see the monographs by Leveque [L92] and Serre [Se99a,Se99b] for more informations on this topic). The main contribution by Crandalland Lions was to find an intrinsic definition of the weak solution which would notrequire to pass to the limit in the regularized problem and could allow to obtainuniqueness results for a large class of stationary and evolutive nonlinear partialdifferential equations.At the very beginning the definition was based on sub- and superdifferentialsand only at a later time the definition based on test functions was introduced([CEL84], see also [Is89]). Starting from these seminal papers the theory had anincreasing success in the following years, as witnessed by the books by Lions [Li82]and Barles [Ba98]. In particular, the latter contains a chapter on discontinuousviscosity solutions. More details on discontinuous viscosity solutions can be foundin the original papers by Barron and Jensen [BJ91], Frankowska [Fr93], Barles[Ba93], and Soravia [Sor93b].The theory has naturally been extended to second order fully nonlinear problems,but the presentation of these results goes beyond the scopes of this book. Forthese extensions of the theory, we refer the reader to the survey paper by Crandall,Ishii and Lions [CIL92] and to the book by Fleming and Soner [FS93].

✐✐42 Chapter 2. Viscosity Solutions of First Order PDEsFinally, it is worth to say that the development of this theory has been largelydriven by the applications to control problems, image processing and fluid dynamics.For the applications to control problems we refer to the monographs by Bardiand Capuzzo Dolcetta [BCD97] for deterministic problems and Fleming and Soner[FS93] for stochastic problems. Moreover, in the books by Sethian [Se96] and Osherand Fedkiw [OF03] it is possible to find find a number of applications of Hamilton–Jacobi equations, in particular in the framework of level set methods.

✐✐Chapter 3Elementary buildingblocksSemi-Lagrangian (SL) schemes require an assembly of different (and partly independent)conceptual blocks, in particular a strategy to move along characteristics anda technique to reconstruct the numerical solution at the foot of a characteristic. Inthe case of HJ equations, a numerical minimization technique is also needed. Here,different choices for all such blocks are considered and explained, and their basictheoretical results are reviewed, with a greater emphasis on less established topics.3.1 A review of ODE approximation schemesThe first typical operations performed by Semi-Lagrangian schemes is to move alongcharacteristics to locate the value to be propagated. This operation amounts toapproximate over a single time step a system of Ordinary Differential Equationslike (1.7), which will be rewritten as⎧⎨ ddt y(x 0, t 0 ; t) = f(y(x 0 , t 0 ; t), t),⎩y(x 0 , t 0 ; t 0 ) = x 0 ,t ∈ R(3.1)with an initial condition x 0 ∈ R d given at the time t 0 , and with a vector fieldf : R d × R → R d globally Lipschitz continuous with respect to its first argument.Such a task can be accomplished by different techniques, usually borrowed fromthe ODE literature. This section will review the basic ideas, which will be laterapplied to the specific situations of interest, whereas for a more extensive treatmentof the related techniques and theoretical results, we refer to specialized textbooks.Throughout the section, we will assume that a set of nodes t k = t 0 + k∆t (withk ∈ Z) is given on the time axis, with y k denoting the corresponding approximationsof y(x 0 , t 0 ; t k ).The usual concepts of consistency and stability are involved in the convergencetheory of methods for ODEs, as the basic convergence theorem states. Conceptually,consistency means that the scheme is a “small perturbation” of the exact equation,whereas stability means that the scheme satisfies (uniformly in ∆t) a principle ofcontinuous dependence upon the initial data. Note that, among the various conceptof stability which have been developed for this problem, we will refer here to zerostability,which is the one required for convergence.43

✐✐44 Chapter 3. Elementary building blocksRather than outlining a general theory, we will sketch how this theory appliesto the two main classes of schemes to approximate (3.1), which are referred to asone-step and multistep methods. We will briefly review their general philosophy,as well as the main results of consistency, stability and convergence, and give somepractical examples, including cases of use in SL schemes.Remark 3.1. While practical schemes for (3.1) are necessarily stable, we will seethat, in the convergence theory of SL schemes, stability is not technically required.At the level of convergence, the choice of a particular scheme rather results in adifferent consistency error, although it may also affect the qualitative behaviour ofthe solution.3.1.1 One-step schemesIn one-step schemes the system (3.1) is discretized in the form{y k+1 = y k + ∆tΦ(∆t; y k , t k , y k+1 )y 0 (x 0 , t 0 ) = x 0 .(3.2)In (3.2), the approximation y k+1 is constructed using only the informations availableat the time step t k . If the function Φ has a genuine dependence on y k+1 , then thescheme is termed as implicit and the computation of y k+1 itself requires to solve(3.2) as a system of nonlinear equations (or a scalar equation if the solution isscalar).ExamplesWe show in Table 3.1 some of the simplest examples of explicit and implicit onestepschemes of first and second order, namely Forward Euler (FE), Backward Euler(BE), Heun (H) and Crank–Nicolson (CN). More in general, this class of schemesincludes all Runge–Kutta type methods. Note that, although their use is clear whenthe dynamics f(·, ·) has an explicit expression, care should be taken when applyingthem to step back along characteristics, especially in nonlinear cases, since theinformation might neither be available at any time step, nor at any point. Thisissue will be discussed in Chapter 5.Theoretical resultsConsistency The scheme (3.2) is said to be consistent with order p ≥ 1 if, oncereplaced y k , y k+1 with the exact values y(t k ) = y(x 0 , t 0 ; t k ), y(t k+1 ) = y(x 0 , t 0 ; t k+1 )of a smooth solution, the scheme satisfiesy(t k+1 ) = y(t k ) + ∆tΦ(∆t; y(t k ), t k , y(t k+1 )) + O(∆t p+1 ). (3.3)In general, we expect that (3.3) would result in obtaining an approximation errorof order O(∆t p ), and this is also true when using (3.2) for approximating characteristicsin SL schemes (in this case, this consistency error affects the term relatedto time discretization alone).

✐✐3.1. A review of ODE approximation schemes 45scheme form orderFE Φ(∆t; y k , t k , y k+1 ) = f(y k , t k ) p = 1BE Φ(∆t; y k , t k , y k+1 ) = f(y k+1 , t k + ∆t) p = 1H Φ(∆t; y k , t k , y k+1 ) = 1 2 [f(y k, t k ) + f(y k + ∆tf(y k , t k ), t k + ∆t)] p = 2CN Φ(∆t; y k , t k , y k+1 ) = 1 2 [f(y k, t k ) + f(y k+1 , t k + ∆t)] p = 2Table 3.1. First- and second-order one-step schemes for the system (3.1)Stability The definition of zero-stability in the case of one-step schemes requiresthat, if (3.2) is used to approximate (3.1) starting from two different initial conditionsx 0 and x ′ 0 (with the corresponding numerical solutions denoted by y k and yk ′ ),then‖y k − y k‖ ′ ≤ C‖x 0 − x ′ 0‖for any k such that t k ∈ [t 0 , t 0 + T ], and with a constant C independent of ∆t.A general result states that if the function Φ is Lipschitz continuous withrespect to its second and fourth argument, the one-step scheme is zero-stable. Notethat all the examples of Table 3.1 satisfy this condition.ConvergenceFinally, we give the convergence theorem for one-step schemes.Theorem 3.2. If the scheme (3.2) is consistent and zero-stable, then, for any ksuch that t k ∈ [t 0 , t 0 + T ],‖y k − y(x 0 , t 0 ; t k )‖ → 0as ∆t → 0. Moreover, if (3.2) is consistent with order p and the solution y issmooth enough, then‖y k − y(x 0 , t 0 ; t k )‖ ≤ C∆t p .3.1.2 Multistep schemesLinear multistep schemes discretize the system (3.1) in the form⎧⎪⎨ y k+1 = ∑ n s−1j=0 α jy k−j + ∆t ∑ n s−1j=−1 β jf(y k−j , t k−j )y 0 = x 0⎪⎩y 1 , . . . , y ns−1 given(3.4)In (3.4), the approximation y k+1 is constructed using the informations available atthe n s time steps t k−ns+1 to t k+1 . The startup values y 1 , . . . , y ns−1 are chosen so

✐✐46 Chapter 3. Elementary building blocksscheme α 0 α 1 β 0 β 1 orderMP 0 1 2 0 p = 2AB2 1 032− 1 2p = 2Table 3.2. Examples of multistep schemes for the system (3.1)as to be an approximation (with suitable accuracy) of the corresponding values ofthe exact solution.As for the case of one-step schemes, if β −1 ≠ 0 so that the right-hand side of (3.4)depends on y k+1 , then the computation of y k+1 requires to solve (3.4) as a systemof nonlinear equations.ExamplesIn table 3.2 we report two specific choices of the coefficients, both of which resultin a method of order p = 2, and corresponds to the midpoint scheme (MP) andto the second-order Adams–Bashforth scheme (AB2). Such cases turn out to be ofcommon use in SL schemes for environmental fluid dynamics.Theoretical resultsConsistency Consistency is checked in multistep schemes according to the sameprinciple used in one-step schemes, namely to replace the values y k with the exactvalues y(t k ) = y(x 0 , t 0 ; t k ) of a smooth solution. Therefore, the multistep scheme(3.4) is said to be consistent with order p ≥ 1 ify(t k+1 ) =n∑s−1j=0n∑s−1α j y(t k−j ) + ∆tj=−1β j f(y(t k−j ), t k−j ) + O(∆t p+1 ). (3.5)It is possible to derive algebraic conditions on the coefficients α j , β j which ensurethat (3.5) is satisfied. However, it is also possible to avoid these technical aspects,and rather work directly on the formulation (3.5).Stability Although similar to the definition adopted for one-step schemes, zerostabilityof multistep schemes is defined in a way which takes into account not onlythe two initial points y 0 = x 0 and y 0 ′ = x ′ 0, but all the startup values y 1 , . . . , y ns−1and y 1, ′ . . . , y n ′ s−1. We require therefore that, for any k such that t k ∈ [t 0 , t 0 + T ],‖y k − y k‖ ′ ≤ C max (‖y i − y ′0≤i≤n s−1i‖)with a constant C independent of ∆t.The endpoint of zero-stability analysis for linear multistep schemes is the socalledroot condition, which gives a necessary and sufficient condition in algebraicform.

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 47Theorem 3.3. A multistep method in the form (3.4) is zero–stable if and only if,once denoted by ζ i (i = 1, . . . , n s ) the roots of the polynomialn∑s−1r(ζ) = ζ ns − α j ζ ns−j−1 ,j=0one has |ζ i | ≤ 1 for any i, and in addition, all roots such that |ζ i | = 1 are simple.Convergence The convergence theorem for multistep schemes, besides the standardassumptions of consistency and stability, requires convergence of the startupvalues to the corresponding values for the solution.Theorem 3.4. If the scheme (3.4) is consistent and zero-stable, and if, for 1 ≤k ≤ n s − 1,‖y k − y(x 0 , t 0 ; t k )‖ → 0,then, for any k such that t k ∈ [t 0 , t 0 + T ],‖y k − y(x 0 , t 0 ; t k )‖ → 0as ∆t → 0. Moreover, if (3.4) is consistent with order p, ‖y k − y(x 0 , t 0 ; t k )‖ =O(∆t p ) for 1 ≤ k ≤ n s − 1 and the solution y is smooth enough, then‖y k − y(x 0 , t 0 ; t k )‖ ≤ C∆t p .3.2 Reconstruction techniques in one and multiplespace dimensionsThe role of the reconstruction step in SL schemes is to recover the value of thenumerical solution at the feet of characteristics, which are not in general grid pointsthemselves. In their basic version, SL schemes do not use cell averages, and the reconstructionis rather based on the interpolation of pointwise values of the solution.We assume therefore that a grid (not necessarily structured) with nodes x j isset, and that a function v(x) is represented by the vector V of its correspondingnodal values v j . The space discretization parameter ∆x, in the basic case of a singlespace dimension with sequentially numbered nodes, is defined as∆x = sup (x k+1 − x k ), (3.6)kwhereas in multiple dimensions and unstructured cases, it requires more complexdefinitions which will be recalled when necessary.In the simplest cases (symmetric Lagrange interpolation, Lagrange finite elements),the reconstruction is set in the form of a linear combination of suitablebasis functions:I[V ](x) = ∑ v k ψ k (x). (3.7)kWe will mostly consider functions ψ k in piecewise polynomial form of degree r, and ifnecessary, the notation I r [V ] will be used to state more explicitly this degree. Also,the ψ k are usually chosen to be cardinal functions, that is, to satisfy ψ k (x i ) = δ ik .

✐✐48 Chapter 3. Elementary building blocksWhen non-oscillatory reconstructions are considered, the linearity with respect tothe values v k is lost and the reconstruction takes a more complex form.Whenever useful, we will apply in the sequel the idea of reference basis functions,meaning a basis of cardinal functions which is defined on a reference grid (e.g.,a grid with unity distance between nodes), and generates the actual basis of thereconstruction by an affine transformation depending on the geometry of the specificgrid used.In the sequel of the section, we will review three reconstruction strategies(symmetric Lagrange interpolation, ENO, WENO) which are usually implementedwith constant step, and extended to multiple dimension by separation of variables.Moreover, a genuinely multidimensional and unstructured strategy (finite elements)will also be considered.3.2.1 Symmetric Lagrange interpolation in R 1This kind of interpolation is usually applied with evenly spaced nodes (by (3.6), ∆xwill denote the constant step), and the samples of V used for the reconstructionare taken from a stencil surrounding the interval which contains the point x, thisresulting in a piecewise polynomial interpolation. Note that, in principle, Lagrangeinterpolation can be implemented on nonuniform grids as well as in globally polynomialform. In the SL framework, however, the former choice would lead in generalto an unstable scheme, the latter to an undue numerical dispersion in the form ofGibb’s oscillations. Lagrange interpolation is typically implemented with an odddegree, thus using an equal number of nodes on both sides of x. In fact, with sucha choice, the interpolation error is related to an even derivative of the function, thisresulting in a symmetric behaviour of the SL scheme around singularities, as it willturn out from the numerical dispersion analysis.ConstructionTaking into account that the number of nodes must be one unity larger then thedegree r of the interpolation, we obtain, for x ∈ [x l , x l+1 ], that the value I[V ](x) iscomputed by a Lagrange polynomial constructed on a set of nodes which includesr+12nodes on each side of the interval. This stencil of nodes will be denoted byS ={x k : l − r − 1 ≤ k ≤ l + r + 122and is depicted in Figure 3.1 for the lowest odd orders of interpolation. As aconsequence, the Lagrange polynomial reconstruction takes the formI r [V ](x) = ∑ ∏ x − x iv k. (3.8)x k − x ix k ∈Sx i∈S\{x k }Since the Lagrange basis used to interpolate changes as x moves from oneinterval to the other, at a first glance is could seem that (3.7) would not be satisfiedwith a unique basis of cardinal functions. Nevertheless, the reconstruction beinglinear with respect to the values to be interpolated, such a unique basis can still besuitably defined. In fact, the interpolation operator I[·] is a linear map from thespace l ∞ of bounded sequences into the space of continuous functions on R:},I[·] : l ∞ → C 0 (R) (3.9)

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 49Figure 3.1. Reconstruction stencil for linear, cubic, quintic Lagrange interpolationso that, by elementary linear algebra arguments, any basis function ψ k can bedefined to be nothing but the image of the base element e k (that is, the interpolationof a sequence such that v k = 1, v i = 0 for i ≠ k).On the other hand, since the grid is uniform and the reconstruction is self-similarand invariant by translation, any such basis function may be written as( ) x − xkψ k (x) = ψ . (3.10)∆xHere, the function ψ plays the role of reference basis function and may be obtainedby applying the interpolation operator to the sequence e 0 , with ∆x = 1. In thesequel, y will represent the variable in the reference space, and we will possiblyuse the notation ψ [r] to denote the reference function for a Lagrange interpolationof order r. Although the interpolation procedure, as performed by (3.8), does notrequire to know the explicit expression of ψ, this expression will be useful later forthe theoretical analysis of the schemes.ExamplesWe sketch below the construction of the Lagrange reconstruction, as well as of thecorresponding reference basis function corresponding to the cases (r = 1, 3, 5) shownabove.Linear (P 1 ) interpolation We first show the simple situation of linear interpolation,which also corresponds to the P 1 case in the finite element setting. Theexpression of the interpolating polynomial for the first order (r = 1) isI 1 [V ](x) = x − x l+1v l +x − x lv l+1 =x l − x l+1 x l+1 − x l= x l+1 − x∆xv l + x − x l∆x v l+1, (3.11)

✐✐50 Chapter 3. Elementary building blockswhereas the interpolation of the sequence e 0 in the reference space results in areference function of the well-known structure⎧⎪⎨ 1 + y if − 1 ≤ y ≤ 0ψ [1] (y) = 1 − y if 0 ≤ y ≤ 1.(3.12)⎪⎩0 elsewhereNote that the value at a certain node affects the reconstruction only in the adjacentintervals.Lagrange interpolation of higher order Take now as a second example cubicreconstruction, which performs an interpolation using the four nearest nodes (twoon the left and two on the right of the interval containing x). The interpolatingpolynomial is computed asI 3 [V ](x) =∑l+2k=l−1v k∏i≠kx − x ix k − x i== − (x − x l)(x − x l+1 )(x − x l+2 )6∆x 3 v l−1 ++ (x − x l−1)(x − x l+1 )(x − x l+2 )2∆x 3v l −− (x − x l−1)(x − x l )(x − x l+2 )2∆x 3 v l+1 ++ (x − x l−1)(x − x l )(x − x l+1 )6∆x 3 v l+2(note that the value at a node x j affects the reconstruction in the interval (x j−2 , x j+2 )).Then, referring again to the case in which e 0 is interpolated and ∆x = 1, the explicitform of the reference base function ψ [3] is⎧12(y + 1)(y − 1)(y − 2) if 0 ≤ y ≤ 1⎪⎨ψ [3] − 1(y) =6(y − 1)(y − 2)(y − 3) if 1 ≤ y ≤ 2.0 if y > 2⎪⎩ψ [3] (−y) if y < 0.(3.13)In turn, a similar computation for the quintic interpolation would yeld for theinterpolant the expression:I 5 [V ](x) =∑l+3k=l−2v k∏i≠kx − x ix k − x i== − (x − x l−1)(x − x l )(x − x l+1 )(x − x l+2 )(x − x l+3 )120∆x 5 v l−2 ++ (x − x l−2)(x − x l )(x − x l+1 )(x − x l+2 )(x − x l+3 )24∆x 5v l−1 −− (x − x l−2)(x − x l−1 )(x − x l+1 )(x − x l+2 )(x − x l+3 )12∆x 5 v l + · · · ++ (x − x l−2)(x − x l−1 )(x − x l )(x − x l+1 )(x − x l+2 )120∆x 5 v l+3 .

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 51Figure 3.2. The basis functions for linear, cubic, and quintic interpolationAccordingly, the reference basis function would be in the form:⎧− 1 12(y + 2)(y + 1)(y − 1)(y − 2)(y − 3) if 0 ≤ y ≤ 11⎪⎨ψ [5] 24(y + 1)(y − 1)(y − 2)(y − 3)(y − 4) if 1 ≤ y ≤ 2(y) = − 1120(y − 1)(y − 2)(y − 3)(y − 4)(y − 5) if 2 ≤ y ≤ 30 if y > 3⎪⎩ψ [5] (−y) if y < 0.(3.14)More in general, for an arbitrary odd interpolation degree r written in the form(3.8), the reference basis function ψ [r] is⎧⎪⎨ψ [r] (y) =⎪⎩[r/2]+1∏k≠0,k=−[r/2].r∏ y − k−kk=1y − k−kif 0 ≤ y ≤ 1if [r/2] ≤ y ≤ [r/2] + 10 if y > [r/2] + 1ψ [r] (−y) if y < 0.(3.15)Note that this construction can also be performed for an even degree of interpolation,but the result would not be a symmetric function.We show in figure 3.2 the reference basis functions for interpolation, ψ [r] (y),for r = 1, 3, 5.

✐✐52 Chapter 3. Elementary building blocksTheoretical resultsWe recall here the approximation result concerning the interpolation error for Lagrangepolynomial approximations. The proof is usually contained in any textbookin basic Numerical Analysis.Theorem 3.5. Let the interpolation I r [V ] be defined by (3.7), with ψ j defined by(3.10) and ψ defined by (3.15). Let v(x) be a uniformly continuous function on R,and V be the vector of its nodal values. Then, for ∆x → 0,If moreover v ∈ W s,∞ (R), then‖v − I r [V ]‖ ∞ → 0. (3.16)‖v − I r [V ]‖ ∞ ≤ C∆x min(s,r+1) (3.17)for some positive constant C depending on the degree of regularity s.Remark 3.6. As we mentioned, globally polynomial interpolations show Gibb’soscillations when treating solutions with singularities, and therefore they are usuallyavoided in nonsmooth contexts (like viscosity or entropy solutions), whereaspiecewise polynomial approximations (e.g., Lagrange and finite element) have a betterperformance. However, although confined to a neighbourhood of the singularity,high-order Lagrange reconstructions can still exhibit an oscillatory behaviour, due tothe changes of sign in the basis functions (see Figure 3.2). This motivates the searchfor non-oscillatory (ENO/WENO) reconstruction strategies which could reduce thisdrawback.3.2.2 Essentially Non-Oscillatory interpolation in R 1Essentially Non-Oscillatory (ENO) interpolation is a first strategy of reconstructionwhich has been developed in order to reduce Gibb’s oscillations. ENO interpolationcuts essentially such oscillations (that is, reduces over- or undershoots of thereconstruction to the order of some power of ∆x), still retaining a high order ofaccuracy in smooth regions of the domain. The basic tool for this operation is astrategy which selects, among all candidate stencils, the stencil where the functionis smoother. In this procedure, different functions may require different stencils, sothat linearity of the reconstruction (expressed by (3.7)) is lost.ConstructionWe define the procedure to construct a reconstruction of ENO type based on theone-dimensional mesh {x j }, in order to approximate the value v(x) for x ∈ [x l , x l+1 ].Most of what follows needs not a uniform mesh, although this is the typical situation.In any case, ∆x will be defined by (3.6).We have to construct an interpolating polynomial of degree r, with a stencilof r + 1 points, which must include x l and x l+1 and be chosen in order to avoidsingularities of v. This will be performed using Newton’s form of the interpolatingpolynomial,I r [V ](x) = V [x l0 ] +r∑k−1∏V [x l0 , . . . , x lk ] (x − x lm ) (3.18)k=1m=0

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 53with the divided differences V [·] defined asV [x l0 ] = v(x l0 ),V [x l0 , . . . , x lk ] = V [x l 1, . . . , x lk ] − V [x l0 , . . . , x lk−1 ]x lk − x l0.Note that, in order to reconstruct v in the interval [x l , x l+1 ], we set x l0 = x l ,x l1 = x l+1 and proceed to extend the stencil by adding further adjacent points.Now, from the the basic results concerning divided differences, it is knownthat, if v ∈ C s (I) (I being the smallest interval containing x l0 , . . . , x ls ), then thereexists a point ξ ∈ I such thatV [x l0 , . . . , x ls ] = v(s) (ξ). (3.19)s!On the other hand, if v has a discontinuity in I, then( ) 1V [x l0 , . . . , x ls ] = O∆x s ,and more in general, if the discontinuity occurs at the level of the k–th derivativev (k) , with k < s, then( ) 1V [x l0 , . . . , x ls ] = O∆x s−k . (3.20)Hence, as long as it affects the construction of the interpolating polynomial, themagnitude of a divided difference is a measure of the regularity of the function onthe related stencil.Let therefore the fixed initial stencil be denoted byS 1 = {x l , x l+1 } = {x l0 , x l1 }. (3.21)At the k–th step of the procedure, we pass from a stencil S k−1 to the stencilS k = {x l0 , . . . , x lk } (3.22)by adding the point x lk . Then, if we plan to extend the stencil towards pointswhere v(x) is smoother, x lk will be chosen (between the two possible choices) as theone which minimizes the magnitude of the new divided difference. More formally,settingl − k = min(l 0, . . . , l k−1 ) − 1, l + k = max(l 0, . . . , l k−1 ) + 1,then:• If |V [x l0 , . . . , x lk−1 , x l−]| < |V [x l0 , . . . , x lk−1 , xkl+ ]|, then l k = l − k kextended one point to the left)(the stencil is• If |V [x l0 , . . . , x lk−1 , x l−]| ≥ |V [x l0 , . . . , x lk−1 , xkl+ ]|, then l k = l + k kextended one point to the right)(the stencil isAfter the selection, the stencil is updated and the algorithm proceeds to thenext step, until k = r. The interpolation may be computed by adding each term ofthe Newton polynomial during the process, or by performing a Lagrange interpolationon the final stencil S = S r .

✐✐54 Chapter 3. Elementary building blocksTheoretical resultsWe review some basic results about ENO interpolation.Accuracy in smooth regions First, is is obvious that if the function to be interpolatedis smooth, then the accuracy coincides with the accuracy of Lagrangeinterpolation:Theorem 3.7. Let the interpolation I r [V ] be defined by (3.18), with the ENOprocedure of selection for the V [x l0 , . . . , x lk ]. Let v ∈ W s,∞ (R). Then,for some positive constant C.‖v − I r [V ]‖ ∞ ≤ C∆x min(s,r+1) (3.23)Bounds on the Total Variation The second property is related to the total variationof the reconstruction. We start with a preliminary result, which does notrequire the ENO procedure for computing I r , but is suitable for any piecewise polynomialinterpolation.Proposition 3.8. Let v be discontinuous in the interval [x l , x l+1 ]. Then, I r [V ] hasno extrema in the same interval.Proof. We only give a sketch of the proof, based on the case of v(x) defined as astep function:{0 x ≤ 0v(x) =1 x > 0.Let p ∈ P r be the polynomial interpolating v on the stencil S = S r , and assume thediscontinuity point x = 0 satisfiesx l < 0 < x l+1with x l , x l+1 ∈ S r . On any interval [x j , x j+1 ] not containing the discontinuity pointwe havep(x j ) = u(x j ) = u(x j+1 ) = p(x j+1 ),and therefore there exists a point ξ j ∈ (x j , x j+1 ) such thatp ′ (ξ j ) = 0.Then, we can find r − 1 different roots for p ′ (x), that is, one root for any interval(x j , x j+1 ) not containing the point x = 0. Since p ′ ∈ P r−1 , it follows that p ′ (x)cannot have further roots in the shock interval [x l , x l+1 ], that is, p(x) is monotonein this interval.Last, we give the result on the total variation of ENO interpolation. Basically,this results states that ENO interpolation is TVB, that is, it has bounded totalvariation. In addition, the increase in variation introduced is of the order of theinterpolation error in smooth regions.

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 55Theorem 3.9. Let v(x) be a piecewise continuous function with derivative v (s)bounded in the continuity set. Then, there exists a function z(x) such thatand satisfyingz(x) = I r [V ](x) + O(∆x min(s,r+1) ),T V (z) ≤ T V (v).Proof. The statement directly follows from the two last results, taking z(x) = v(x)in intervals where v is smooth, and z(x) = p(x) in interval containing discontinuities.3.2.3 Weighted Essentially Non-Oscillatory interpolation in R 1WENO interpolation is another strategy for non-oscillatory reconstruction. It isderived from the ENO strategy, but unlike ENO it uses all the computed informationas much as possible.In practice, while the ENO strategy computes twice the required number ofdivided differences to build a Newton polynomial, but at each step one differenceamong two is discarded, WENO strategy computes all the candidate interpolatingpolynomials. Then, if the function is smooth enough on the given stencil, candidatepolynomials are combined in order to have a further increase of the accuracy. If thefunction is nonsmooth, then the reconstruction selects only the smoothest candidateapproximation.Although the construction of a WENO interpolation is a well-established matter,we will review here the details of the procedure, along with the main theoreticalresults, since it is usually treated and used in a different framework. In this part,we do not necessarily assume a uniform mesh spacing.ConstructionGeneral structure To construct a WENO interpolation of degree r = 2n − 1 onthe interval [x l , x l+1 ], we start from the Lagrange polynomial built on the stencilS = {x l−n+1 , . . . , x l+n }, and written in the formn∑Q(x) = C k (x)P k (x) (3.24)k=1where the “linear weights” C k are polynomials of degree n − 1 and the P k arepolynomials of degree n interpolating V on the stencil S k = {x l−n+k , . . . , x l+k },k = 1, . . . , n (note that all the stencils S k overlap on the interval [x l , x l+1 ], and thatthe dependence on l has been dropped in this general expression). The nonlinearweights are then constructed so as to obtain an approximation of the highest degreeif suitable smoothness indicators β k give the same result on all stencils. This leadsto defineα k (x) =C k(x)(β k + ε) 2 (3.25)(with ε a properly small parameter, usually of the order of 10 −6 ), and then thenonlinear weights asw k (x) =α k(x)∑h α h(x) . (3.26)

✐✐56 Chapter 3. Elementary building blocksThe final form of WENO interpolation is thenI r [V ](x) =n∑w k (x)P k (x), (3.27)k=1where w k is defined by (3.25)–(3.26).The general concept of (3.27) is the following. If all the smoothness indicatorsβ k would have the same value, then w k (x) = C k (x) and the reconstruction is givenby the Lagrange polynomial (3.24), which has the highest possible degree. If someof the smoothness indicators has a “large” value with respect to the others (thismeaning that the partial stencil contains a singularity), then w k (x) ≪ C k (x) andtherefore the corresponding term is discarded via the weighting process.Smoothness indicators Some degrees of freedom are left in the choice of thesmoothness indicators. The general idea is that in smooth regions all indicatorsβ k should basically estimate the same quantity, and vanish at a certain rate when∆x → 0, that isβ k = D j ∆x 2p (1 + O(∆x q )) (3.28)where D j is some constant depending on the function and on the interval index j. Ifcondition (3.28) is satisfied, then in smooth conditions we would have w k (x) ≈ C k (x)(in a form depending on the exponent q, to be made more precise later), and thisenables that scheme to increase the accuracy by merging the information fromdifferent stencils. On the other hand, if β k is associated to a stencil containing adiscontinuity of the function, then we require thatβ k = O(1) (3.29)so that asymptotically such a stencil would be discarded by the weighting procedure.Following [Sh98], a typical way of defining the smoothness indicators is basedon successive derivatives P (h)kof the polynomials P k , and more preciselyβ k =n∑h=1∫ xl+1for which (3.28), (3.29) hold with p = 1, q = 2.x l∆x 2h−1 P (h)k(x) 2 dx, (3.30)Linear weights Let us consider now the linear weights C k , which have not yet beengiven a precise expression. Due to (3.24) and to the definition of the polynomialsP k , the C k are characterized by the fact that Q(x) should be the interpolatingpolynomial of V (x) on the stencil S. Since P k interpolates V on the stencil S k , itwould be natural to require that C k should vanish at the nodes outside S k , and thatin the nodes of S the nonzero weights should have unit sum. In this way, if P k isthe polynomial that interpolates the function on S k , then (3.24) is the polynomialthat interpolates V on S.Taking into account that the number of nodes belonging to S but not to S kis precisely n, we thus infer that the linear weights must necessarily have the form∏C k (x) = γ k (x − x h ) = γ k ˜Ck (x) (3.31)x h ∈S\S k

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 57with γ k to be determined in order to have unit sum. Note that the conditionsn∑C k (x i ) = 1 (3.32)k=1for each x i ∈ S apparently constitute a set of 2n equations in n unknowns. Actually,the correct perspective to look at these conditions is the following. The left-handside of Equation (3.32) is a polynomial of degree n−1 (computed at x i ), and on theright-hand side we have the polynomial p(x) ≡ 1. Therefore, imposing that the twopolynomials coincide on more than n − 1 points is equivalent to impose that theyare identical. Now, it is easy to show that the monic polynomials ˜C k , k = 1, . . . , n,are independent and form a basis of the space P n−1 of polynomials of degree n − 1.Conditionn∑C k (x) = 1 ∀x ∈ R (3.33)k=1uniquely determines the constants γ k as a polynomial identity.Condition (3.33) can be conveniently satisfied by imposing it on a suitable setof points. Consider first the node x l−n+1 . Then, using (3.31) into (3.24), we notethat, among all the linear weights computed at x l−n+1 , the only nonzero weight isC 1 , so thatQ(x l−n+1 ) = C 1 (x l−n+1 )P 1 (x l−n+1 ) = C 1 (x l−n+1 )v l−n+1 (3.34)and this implies that C 1 (x l−n+1 ) = 1 and allows to compute γ 1 . On the other hand,in the following node x l−n+2 , the nonzero weights are C 1 and C 2 , and with a similarargument we obtain the conditionQ(x l−n+2 ) = C 1 (x l−n+2 )P 1 (x l−n+2 ) + C 2 (x l−n+2 )P 2 (x l−n+2 ) == [C 1 (x l−n+2 ) + C 2 (x l−n+2 )]v l−n+2 , (3.35)which implies that C 1 (x l−n+2 ) + C 2 (x l−n+2 ) = 1, whence γ 2 can be computed.Following this guideline, we finally obtain the set of conditionsk∑C i (x l−n+k ) = 1 (k = 1, . . . , n), (3.36)i=1that is, more explicitly,k∑i=1γ i∏x h ∈S\S i(x l−n+k − x h ) = 1 (k = 1, . . . , n), (3.37)which is a linear triangular system in the unknowns γ i .ExamplesWe give two examples of this construction (on a uniform mesh), including theexpressions for the smoothness indicators (3.30). A general procedure to computelinear weights will be given in the next subsection.

✐✐58 Chapter 3. Elementary building blocksSecond-third order WENO interpolation To construct a third order interpolationwe start from two polynomials of second degree, so thatI 3 [V ](x) = w L P L (x) + w R P R (x), (3.38)where P L (x) and P R (x) are second-order polynomials constructed respectively onthe nodes x l−1 , x l , x l+1 and on the nodes x l , x l+1 , x l+2 . The two linear weightsC L and C R are first degree polynomials in x, and according to the general theoryoutlined so far, they readC L = x l+2 − x3∆x , C R = x − x l−13∆x , (3.39)and the expressions of α L , α R , w L and w R may be easily recovered from the generalform.According to (3.30), the smoothness indicators have the explicit expressionsβ L = 1312 v2 l−1 + 16 3 v2 l + 2512 v2 l+1 − 13 3 v l−1v l + 7 6 v l−1v l+1 − 19 3 v lv l+1 , (3.40)β R = 1312 v2 l+2 + 163 v2 l+1 + 2512 v2 l − 13 3 v l+2v l+1 + 7 6 v l+2v l − 19 3 v lv l+1 . (3.41)Third-fifth order WENO interpolation To construct a fifth order interpolationwe start from three polynomials of third degree:I 5 [V ](x) = w L P L (x) + w C P C (x) + w R P R (x), (3.42)where the third-order polynomials P L (x), P C (x) and P R (x) are constructed respectivelyon x l−2 , x l−1 , x l , x l+1 , on x l−1 , x l , x l+1 , x l+2 and on x l , x l+1 , x l+2 , x l+3 . Theweights C L , C C and C R are second degree polynomials in x, and have the formC L = (x − x l+2)(x − x l+3 )20∆x 2 ,C C = (x − x l−2)(x − x l+3 )10∆x 2 ,C R = (x − x l−2)(x − x l−1 )20∆x 2 ,while the smoothness indicators β C and β R have the expressionsβ C = 6145 v2 l−1 + 33130 v2 l + 33130 v2 l+1 + 6145 v2 l+2 − 14120 v l−1v l + 17930 v l−1v l+1 ++ 293180 v l−1v l+2 − 125960 v lv l+1 + 17930 v lv l+2 − 14120 v l+1v l+2 , (3.43)β R = 40790 v2 l + 72130 v2 l+1 + 24815 v2 l+2 + 6145 v2 l+3 − 119360 v lv l+1 + 43930 v lv l+2 −− 683180 v lv l+3 − 230960 v l+1v l+2 + 30930 v l+1v l+3 − 55360 v l+2v l+3 , (3.44)and β L can be obtained using the same set of coefficients of β R in a symmetric way(that is, replacing the indices l − 2, . . . , l + 3 with l + 3, . . . , l − 2).

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 59Theoretical resultsIt is easy to check that, due to their structure, the linear weights for the twoexamples above are always positive in the interval [x l , x l+1 ]. We prove now thatthis result holds in general. In addition, we give an explicit expression for thelinear weights in the case of evenly spaced nodes, and prove an interpolation errorestimate.Positivity of the weights We first give a general proof of positivity for the linearweights C k (x). We point out that, due to the general structure of nonlinear weights,positivity of the former set of weights implies positivity of the latter.Theorem 3.10. Let {x i } be a family of consecutively numbered points of R. Then,once the Lagrange polynomial Q(x) of a given function v(x) built on the stencilS = {x l−n+1 , . . . , x l+n } is written in the form (3.24), the linear weights C k (x)(k = 1, . . . , n) are nonnegative for any x ∈ (x l , x l+1 ). Moreover, ∑ k C k(x) ≡ 1.Proof. Let us denote by Q (h,m) (x) (with j −n+1 ≤ h ≤ j < j +1 ≤ m ≤ j +n, andm − h ≥ n) the interpolating polynomial constructed on the stencil {x h , . . . , x m },so thatQ(x) = Q (j−n+1,j+n) (x), (3.45)and moreoverWe will also write a generic Q (h,m) (x) asP k (x) = Q (j−n+k,j+k) (x). (3.46)Q (h,m) (x) = ∑ kC (h,m)k(x)P k (x), (3.47)where the summation may be extended to all k = 1, . . . , n by setting C (h,m)k= 0whenever the stencil of P k is not included in {x h , . . . , x m }. Therefore, the finallinear weights will beC k (x) = C (j−n+1,j+n)k(x). (3.48)We proceed by induction on the degree of Q (h,m) , given by i = deg Q (h,m) =m − h, starting with i = n up to i = 2n − 1. First we prove that the claim ofthe Theorem extends to all polynomials Q (h,m) (x) defined in (3.47). The claimis obviously true for Q (j−n+k,j+k) (x) by (3.46). In this case the coefficients readC (h,m)k= 1, C s(h,m) ≡ 0, s ≠ k. Then, we assume the claim is true for the set oflinear weights C (h,m)k(x) of a generic Q (h,m) (x) such that m − h = i and add a nodeto the right (adding a node to the left leads to an analogous computation). Byinductive assumption we have, for any h and m such that h ≤ j < j + 1 ≤ m andm − h = i, that (3.47) holds with C (h,m)k(x) ≥ 0. On the other hand, by elementaryinterpolation theory arguments (Neville’s recursive form of the interpolatingpolynomial),Q (h,m+1) (x) = x m+1 − xQ (h,m) (x) +x − x hQ (h+1,m+1) (x). (3.49)x m+1 − x h x m+1 − x hNote that deg Q (h,m) = deg Q (h+1,m+1) = i, and that both fractions which multiplyQ (h,m) (x) and Q (h+1,m+1) (x) are positive as long as x ∈ (x l , x l+1 ). Therefore, using

✐✐60 Chapter 3. Elementary building blocks(3.47) into (3.49), we getQ (h,m+1) (x) = ∑ [xm+1 − xC (h,m)k(x) + x − x ]hC (h+1,m+1)k(x) P k (x).x m+1 − x h x m+1 − x hk(3.50)The term in square brackets is C (h,m+1)k(x) and it is immediate to check that it isnonnegative. Iterating this argument up to Q (j−n+1,j+n) (x) completes the proof ofnonnegativity for all the linear weights C k (x).Lastly, setting v(x) ≡ 1, we also have Q(x) ≡ 1, and, for any k, P k (x) ≡ 1.Plugging this identities into (3.24) we obtain ∑ k C k(x) ≡ 1.Explicit form of the linear weights We turn now to the problem of giving anexplicit expression to the linear weights in the situation of evenly spaced nodes.The expression of the weights for the linear scheme is∏C k = γ kx h ∈S\S k(x − x h ),so that the problem is to provide the expression of the constants γ k by applying tothe case of a uniform grid the general procedure already outlined. The endpoint ofthis analysis is given by the following theorem.Theorem 3.11. Assume the space grid is evenly spaced with step ∆x. Then, thelinear weights (3.31) have the explicit formC k (x) = (−1)n+k∆x n−1 n!(n − 1)!(n − k)!(k − 1)!(2n − 1)!∏x h ∈S\S k(x − x h ) (3.51)and satisfy the positivity condition C k (x) > 0 for any k ∈ {1, . . . , n}, x ∈ [x l , x l+1 ].Proof. First observe that the polynomial C k (x) does not change sign in the interval[x l , x l+1 ]. More precisely, the sign of it, in this interval, is given by sgn(γ k )(−1) n+k .ThereforeC k (x) > 0 ∀x ∈ [x l , x l+1 ], ∀k, n ⇔ sgn(γ k ) = (−1) n+k . (3.52)For simplicity, set l = n in the definition of the stencil S, so that S = {x 1 , . . . , x 2n }.Let us introduce the reference stencils˜S = S/∆x, ˜Sk = S k /∆x.Then the polynomials C k can be written as∏C k (∆xη) = ˜γ k (η − h),h∈ ˜S\ ˜S kwhere η is the variable in the reference space, ˜γ k = γ k ∆x n−1 . The dimensionlessconstants ˜γ k satisfy the conditionk∑˜γ ii=12n∏h=1h∉{i,...,i+n}(i − h) = 1.

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 61Introducing now the matrixa ki =∏ 2nh=1∏ i+nh=ithe system can be written as(k − h)=(k − h)(k − 1)!(2n − k)!(n + i − k)!(k − i)! (−1)n+i , (3.53)k∑a ki˜γ i = 1.i=1Let µ i ≡ (−1) n+i˜γ i . Then, such constants satisfy the triangular systemk∑|a ki |µ i = 1 (3.54)i=1and according to (3.52) C i (x) > 0, x ∈ [x n , x n+1 ] if and only if µ i > 0.The solution to system (3.54) can be explicitly given. In fact, we will provenow thatµ i =n!(n − 1)!(n − i)!(i − 1)!(2n − 1)!(i = 1, . . . , n). (3.55)Actually, plugging (3.55) and (3.53) into (3.54) we obtain the set of conditionsthat we want to check:k∑ (k − 1)!(2n − k)!n!(n − 1)!(n + i − k)!(k − i)!(n − i)!(i − 1)!(2n − 1)! = 1.i=1These relations can be rearranged in the form(k − 1)!(2n − k)!(2n − 1)!j=0k∑i=1( ) ( )n n − 1k − i i − 1and hence, rewriting the first term and shifting the summation index as j = i − 1,k−1∑( ) ( ) ( )n n − 1 2n − 1= ,k − 1 − j j k − 1which is in turn a special case of the identity (see [AS64])N∑j=0with N = k − 1, and m = n − 1.( ( ) n m=N − j)j( ) n + m,N= 1,Remark 3.12. We can further prove here that the µ i are inverse of integers. Infact, note first that by (3.55), µ i = µ n−i+1 , so that it suffices to prove the claim fori ≤ (n + 1)/2. Then, we rewrite the µ i as⎧n!if i = 1⎪⎨ (2n − 1)!µ i =⎪⎩(n − i + 1) · · · (n − 1)n(n − 1)!(i − 1)!(2n − 1)!if i > 1.(3.56)

✐✐62 Chapter 3. Elementary building blocksThe claim is obvious if i = 1, whereas for i > 1 we derive from (3.56)µ i =(2n − 2i + 2) · · · (2n − 2)2 i−1 (i − 1)!(n + 1) · · · (2n − 1) .Now, if i ≤ (n+1)/2, then 2n−2i+2 ≥ n+1 and therefore any term of the product(2n−2i+2) · · · (2n−2) can be simplified with a term of the product (n+1) · · · (2n−1).This completes the proof of the claim.We report in the following table the value of 1/µ i , for values of n up to n = 8.n 1/µ i1 12 3, 33 20, 10, 204 210, 70, 70, 2105 3024, 756, 504, 756, 30246 55440, 11088, 5544, 5544, 11088, 554407 1235520, 205920, 82368, 61776, 82368, 205920, 12355208 32432400, 4633200, 1544400, 926640, 926640, 1544400, 4633200, 32432400Table 3.3. Values of 1/µ i for the first 8 values of nInterpolation error estimates We finally prove here the convergence result forWENO interpolations, restricting to the smooth case.Theorem 3.13. Let v(x) be a C ∞ function, and the interpolation I r [V ] be definedby the WENO form (3.27) with the smoothness indicators β k satisfying (3.28).Then,‖v − I r [V ]‖ ∞ ≤ C∆x min(2n,n+q+1) . (3.57)Proof. First, note that if (3.28) is satisfied, thenw k (x) = C k (x) + ω k (x, ∆x q ) (3.58)where for any k, ω k (x, ∆x q ) = O(∆x q ), and, since both the w k and the C k haveunity sum, we also have ∑ k ω k(x, ∆x q ) = 0. Then,|v(x) − I r [V ](x)| ≤∣ v(x) − ∑ k∣ ∣∣∣∣ C k (x)P k (x)∣ + ∑C k (x)P k (x) − ∑kkw k (x)P k (x)∣ .For the first term it is clear that∣ v(x) − ∑ kC k (x)P k (x)∣ ≤ C∆x2n , (3.59)

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 63whereas for the second, using (3.58) and the fact that the P k are themselves interpolatingpolynomials of degree n,∑C k (x)P k (x) − ∑ ∣ ∣∣∣∣ w k (x)P k (x)∣∣ = ∑[C k (x) − w k (x)]P k (x)∣ =kkk∑=ω k (x, ∆x q ) [ v(x) + O(∆x n+1 ) ]∣ ∣ ∣∣∣=∣k=∣ v(x) ∑ ω k (x, ∆x q ) + O(∆x n+q+1 )∣ =k= O(∆x n+q+1 ) (3.60)where in the last display we have used the fact that the ω k (x, ∆x q ) have zero sum.Finally, (3.57) follows from (3.59), (3.60).3.2.4 Handling multiple dimensions by separation of variablesLagrange interpolation When passing to multiple space dimensions, the extensionof Lagrange interpolation (that is, in practice, the definition of suitable cardinalbasis functions) can be performed in different forms. In the simplest case, cardinalbasis functions are defined by product of one-dimensional functions, in the formthat follows. Let the space grid of points x j be uniform and orthogonal, and j =(j 1 , . . . , j d ) be the multi-index associated to a given node. Now, once ensured thatthe ψ j are cardinal functions, the form (3.7) still defines an interpolant of v. On theother hand, on a uniform orthogonal grid, a basis of Lagrange cardinal functionscan be defined byd∏ψ j (ξ) = ψ jk (ξ k ). (3.61)k=1where, in order to avoid ambiguities, we have denoted the space variable by ξ =(ξ 1 , . . . , ξ d ). In (3.61), it should be understood that ψ j (which is indexed by amulti-index and has an argument in R d ) refers to the d–dimensional grid, whereasψ jk (which has a simple index, and has a scalar argument) refers to the grid on thek–th variable. It is clear that (3.61) defines a cardinal function, and in fact{1 if i 1 = j 1 , . . . , i d = j dψ j (x i ) =0 else.Figure 3.3 shows the bidimensional version of the cubic interpolation reference basisfunction of Figure 3.2.A different interpretation of this procedure can be given by splitting the operationimplied by (3.7). Working for simplicity in two dimensions, and plugging(3.61) into (3.7), we haveI[V ](ξ) = ∑ ∑v(x j1,j 2)ψ j1 (ξ 1 )ψ j2 (ξ 2 ) =j 1 j 2⎡⎤= ∑ ⎣ ∑ v(x j1,j 2)ψ j1 (ξ 1 ) ⎦ ψ j2 (ξ 2 ). (3.62)j 2 j 1

✐✐64 Chapter 3. Elementary building blocks2.01.51.0Z0.50.0!0.52.52.01.51.0Y0.0!1.0!2.0!2.0!1.00.0X1.01.52.02.5Figure 3.3. Reference basis function for cubic interpolation in R 2In practice, the process appears to be the result of two (in general, d) nested onedimensionalinterpolations. The upper diagram of Figure 3.4 shows the pointsinvolved in this process for a linear Lagrange interpolation. First, the point ξ is locatedin a rectangle of the grid, say [l 1 ∆x 1 , (l 1 +1)∆x 1 ]×[l 2 ∆x 2 , (l 2 +1)∆x 2 ]. Then,in the first phase of the interpolation, two one-dimensional linear reconstructionsare performed to recover the interpolated values at (ξ 1 , l 2 ∆x 2 ) and (ξ 1 , (l 2 +1)∆x 2 ).In the second phase, a linear interpolation is performed between these two valuesto recover the interpolated value at (ξ 1 , ξ 2 ).Once arranged the same points in a tree (as shown in the lower diagram),the interpolation is computed by advancing from the leaves towards the root (interms of data structures, the interpolated value is computed as the tree is visitedin postorder).In general, this procedure can be applied to any dimension d, as well as anyorder of interpolation. The dimension appears as the depth of the tree, whereasthe number of nodes required for one-dimensional reconstruction coincides with thenumber of sons of a given node of the tree.ENO/WENO interpolation The nonlinear nature of non-oscillatory reconstructionsprevents them to be extended to R d in the form (3.7), (3.61). On the otherhand, the mechanism of successive one-dimensional interpolations may also be appliedto ENO or WENO (or in general, to nonlinear) reconstructions, although inthis case, due to the existence of various candidate stencils, the total number ofpoints involved is considerably higher. It is also possible, however, to define nonoscillatoryinterpolants based on genuinely multidimensional stencils, possibly onunstructured grids.3.2.5 Finite element interpolationFinite element interpolation is a reconstruction strategy which, still interpolating agiven function in a piecewise polynomial form, does not need any particular structureof the space grid, and is intrinsically multidimensional, although it also includesthe one-dimensional case.

✐✐// a circle with ceborder C0(t=0,2*pi) { x = 5 * cos(t); y = 5 * sin(t)border C1(t=0,2*pi) { x = 2+0.3 * cos(t); y = 3*sin(border C2(t=0,2*pi) { x = -2+0.3 * cos(t); y = 3*sin3.2. Reconstruction techniques in one and multiple space dimensions 65mesh Th = buildmesh(C0(60)+C1(-50)+C2(-50));plot(Th,ps="electroMesh");fespace Vh(Th,P1);Vh uh,vh; //problem Electro(uh,vh) = //int2d(Th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) )+ on(C0,uh=0) //+ on(C1,uh=1)+ on(C2,uh=-1) ;Electro; // solve the problem, seeplot(uh,ps="electro.eps",wait=true);Figure 3.4. Dimensional splitting for linear interpolation in R 2Figure 9.3: Disk with two elliptical holesFigure 3.5. Tessellation of a complex geometry into triangles, includingmesh adaptationFigure 9.4: Ecated in right hA definite advantage of finite element interpolations is the ability to treatcomplex geometries, which in general do not allow for a structured grid. We showin Figure 3.5 an example, which also illustrates the possibility of performing localrefinements for accuracy purposes.

✐✐66 Chapter 3. Elementary building blocksConstructionBy definition, a finite element is a triple (K, Σ, V r ), where• K is the reference domain of the element, and also in practice the kind ofelementary figure into which the computational domain is split. In one spacedimension K is necessarily an interval; in R 2 typical choices are triangles, andless frequently rectangles, and so on;• V r is the space of polynomial in which the interpolation is constructed. Twotypical choices we will consider here are P r (the space of polynomials in dvariables of degree no larger than r) and Q r (the space of polynomials ofdegree no larger than r with respect to each variable);• Σ is a set of functionals γ i : P r → R (usually referred to as degrees of freedom)which completely determine a polynomial of P r . In the case of Lagrangefinite elements, which is the one of interest here, the degrees of freedom arethe values of the polynomial on a unisolvent set of nodes in K (with someabuse of notation, they are identified with the nodes), and the polynomial isexpressed in the basis of Lagrange functions associated to such set of nodes.We will not explain in detail the construction of a Lagrange finite elementinterpolation in its greatest generality (this topic is treated in the specialized literature),but rather sketch the basic ideas by working on some one- and twodimensionalexamples. As a starting point, the computational domain should be(approximately) tessellated with non-overlapping elements obtained by affine transformationsof the reference element. Interpolation nodes are placed on each elementby mapping the nodes of the reference element. As outlined for symmetric Lagrangeinterpolation, the basis function ψ k used in (3.7) can be conceptually obtained byinterpolating the sequence e k .ExamplesTo make the main ideas clear, we present here the construction of the P 1 , P 2 , Q 1and Q 2 Lagrange finite element spaces in one and two space dimensions. For ageneral treatment of the topic, we refer the reader to classical monographs on finiteelement schemes.P 1 and P 2 finite elements in R In one dimension, the only possibility is to definethe reference element as an interval. In order to interpolate with a P 1 or P 2 basis,respectively two and three nodes must be placed in the reference interval. To obtaina continuous interpolant, two of them must be placed at the interface betweenelements, that is, in the extreme points of the reference element. Therefore, if thereference interval keeps unity distance between nodes, in the P 1 case it could bedefined as the interval [0, 1] with the nodes η 0 = 0, η 1 = 1 (here and in the sequel,we use η as a variable in the reference space). As a result, the Lagrange referencebasis functions (shown in the left plot of Figure 3.6) will be given byl 0 (η) = 1 − η, l 1 (η) = η,while a generic point x ∈ [x l , x l+1 ] will be carried in the reference interval [0, 1] bythe transformationη =x − x l.x l+1 − x l

✐✐3.2. Reconstruction techniques in one and multiple space dimensions 67The cardinal basis function ψ k associated to a node x k of a one-dimensional gridreads then:⎧x − x k−1if x ∈ (x k−1 , x k ]x k − x k−1⎪⎨ψ k (x) = 1 − x − x kif x ∈ (x k , x k+1 )x k+1 − x k⎪⎩0 elsewhereNote that, unless for allowing a nonuniform grid, P 1 finite element interpolationcoincides with first-order Lagrange interpolation. This correspondence, however, islost for higher interpolation orders.In the case of P 2 interpolation, we can choose the reference interval as [0, 2] andplace the three nodes at η = 0, 1, 2. The three Lagrange reference basis functions(shown in the right plot of Figure 3.6) are now given byl 0 (η) = 1 2 (η − 1)(η − 2), l 1(η) = 1 − (η − 1) 2 , l 2 (η) = 1 η(η − 1),2and, if x ∈ [x l , x l+2 ], the transformation from the variable x to the reference variableη isη = 2x − x l.x l+2 − x lThe cardinal basis function ψ k is constructed in a way which parallels the P 1 case,in particular⎧ ( ) 2 x − xk−1⎪⎨ 1 −− 1 if x ∈ (x k−1 , x k+1 )x k − x k−1ψ k (x) =⎪⎩0 elsewhereif x k is internal to an element, and⎧12⎪⎨ψ k (x) =12( x − xk−2x k − x k−1) ( x − xk−2x k − x k−1− 1( ) ( )x − xkx − xk− 1− 2x k+1 − x k x k+1 − x k)if x ∈ (x k−2 , x k ]if x ∈ (x k , x k+2 )⎪⎩0 elsewhereif x k is at the interface between two elements.Note that in the P 2 case the situations in which x k is internal or extreme ofan element leads to different forms, even on an uniformly spaced mesh. More ingeneral, in the P r case (with r > 1), the form of ψ k depends on the position of x kwithin the element. Thus, in any of these cases the interpolation fails in general tobe translation invariant.We show in Figure 3.6 two examples of reference intervals and basis functionsfor P 1 and P 2 finite element interpolation in one space dimension.

✐✐68 Chapter 3. Elementary building blocks1.21.2110.80.80.60.60.40.40.20.200-0.2-0.2-0.40 0.2 0.4 0.6 0.8 1-0.40 0.5 1 1.5 2Figure 3.6. P 1 and P 2 reference elements and basis functions in RP 1 , P 2 , Q 1 and Q 2 finite elements in R 2 At the increase of dimensions, thegeometry of the reference element allows for an increasing number of options. InR 2 , two main situations of interest can be recognized: the situations in which thereference element is respectively a triangle (leading to P r finite element spaces) anda rectangle (leading to Q r f.e. spaces).On triangular elements, it is necessary to construct the space of polynomialsof degree r, with r = 1, 2 in our example. We will not give the explicit expressionof the reference basis functions, but rather show the choice of the nodes. In the P 1case, the polynomial space has dimension three, and the nodes are placed in thevertices of the element. In the P 2 case, the space has dimension six, and the nodesare typically placed at vertices and midpoints. Upper line of Figure 3.7 shows theposition of nodes in the two situations. Clearly, the k–th Lagrange reference basisfunction is a polynomial of degree r taking the value l k (η m ) = δ km at a genericnode η m . Note that the restriction of a Lagrange basis function to one side of thetriangle gives a polynomial of degree respectively r = 1 and r = 2, and the numberof nodes on the side is enough to determine univocally this polynomial. Thus, theinterpolation remains continuous passing from one element to the other.On quadrilateral elements, the Lagrange polynomial are constructed as tensorproducts of one-dimensional Lagrange bases, as it has been discussed concerningmultidimensional Lagrange reconstruction. Therefore, the Q 1 finite element hasfour nodes at the vertices of the quadrilateral, whereas the Q 2 finite element hasnine nodes (vertices, midpoints and center of the quadrilateral element). The lowerline of Figure 3.7 reports the position of nodes in this case. Again, the number ofnodes on each side of the quadrilateral implies that the interpolation is continuousat the interface of two elements.Theoretical resultsThe approximation result concerning finite element interpolation follows the generalresult for polynomial approximations. However, when dealing with nonuniformspace grids, care should be taken to avoid that triangular elements could degeneratewhen refining the grid, typically by flattening towards segments. While we leave aformal theory to specialized literature, we simply remark here that grid refinementin a finite element setting requires that the largest size ∆x of the elements vanishes,and that the grid is nondegenerate. This means that, once denoted by R K theradius of the element K and by r K the radius of the inscribed circle, their ratio

✐✐3.3. Function minimization 69Figure 3.7. P 1 –P 2 (upper) and Q 1 –Q 2 (lower) reference elements in R 2remains bounded for all K:R K≤ C. (3.63)r KLast, we give the approximation result for finite element interpolations.Theorem 3.14. Let the interpolation I r [V ] be defined by a P r finite element spaceon a space grid satisfying (3.63). Let v(x) be a uniformly continuous function onR, and V be the vector of its nodal values. Then, for ∆x → 0,If moreover v ∈ W s,∞ (R), then‖v − I r [V ]‖ ∞ → 0. (3.64)‖v − I r [V ]‖ ∞ ≤ C∆x min(s,r+1) (3.65)for some positive constant C depending on the degree of regularity s.3.3 Function minimizationAs a building block of Semi-Lagrangian schemes for nonlinear problems, we will beconcerned with the problem of minimization of a function of N variables:f(x ∗ ) = minx∈R N f(x),and with the related numerical techniques. Since in our case the function to beminimized has no explicit form, our interest here is mainly focused on derivativefreemethods, i.e. methods which minimize a function without making use of the

✐✐70 Chapter 3. Elementary building blocksFigure 3.8. Updating of a simplex in the basic version of simplex methodanalytic expressions of gradient or hessian. Without aiming at completeness, a listof the main approaches used to perform this operation includes:• Direct search methods• Descent methods (adapted to work without derivatives)• Powell’s method and its modifications• Trust-region methods based on quadratic models.We will briefly review the general philosophy of the various classes of methods.The huge amount of related studies cannot be condensed in such a brief survey;rather than giving implementation details, we will sketch the general ideas andrefer the reader to suitable, specialized literature and software packages.3.3.1 Direct search methodsThe name of direct search methods is generally used to collect methods of heterogeneousnature, which share the feature of minimizing a function without using, eitherin exact or approximate form, its gradient. Direct search methods rather proceedby computing the function at a discrete, countable set of points.The most common schemes of this class are based on the simplex technique(not to be confused with the algorithm used in Linear Programming), whose basicidea dates back to the ’60s and is shown in Figure 3.8.Let a simplex of vertices V 1 , . . . , V m be constructed in R N . Once computedthe corresponding values y 1 , . . . , y m of f at the vertices, the highest value y h isselected, and the corresponding vertex V h is reflected with respect to the centroidC of the simplex made by the remaining vertices. Then, the new simplex obtainedby replacing the node V h with V r is accepted if the value of the function at V r is notthe highest value in the new simplex. Otherwise, the second highest value of theoriginal simplex is selected, and the procedure is repeated. If all possible updateof the simplex lead to find the highest value on the reflected node, the algorithmstops. The size of the simplex should then be considered as a tolerance in locatingthe minimum, and to refine the search the algorithm should be restarted with asimplex of smaller size.

✐✐3.3. Function minimization 71In a later (and more widely used) version, the algorithm has been restatedreplacing the reflection of V h with a discrete line search. In this version the geometryof simplices may change, and while this allows the scheme to adapt to thegeometry of the function, it can also cause the simplices to degenerate, thus stallingthe algorithm. Specific tools have been devised to handle this situation, the mostobvious one being the restart.3.3.2 Descent methodsIn descent methods, an initial estimate of the minimum point x 0 is given, and theupdating of the approximation x k is performed via the formulax k+1 = x k + β k d k , (3.66)that is, the scheme moves from x k to x k+1 along the direction d k with step β k . Asa rule (which also gives the name to this class of schemes), the direction d k shouldbe a descent direction, i.e. (d k , ∇f(x k )) < 0, β k should be positive, and f(x k+1 )

✐✐72 Chapter 3. Elementary building blocksFigure 3.9. Construction of conjugate directions in Powell’s methodthey are based on search directions d k which are A–conjugate, that is,(Ad i , d i ) > 0, (Ad i , d j ) = 0 (i ≠ j).In the case of quadratic functions, it can be proved that such a scheme convergesat most in n steps. Its adaptation to nonquadratic function (the socalledconjugate gradient, which requires to compute ∇f at each iteration)has superlinear convergence, while the cost per iteration is O(N 2 ).To turn back to the problem of implementing schemes of this class withoutcomputing derivatives, we remark that in principle it is possible to replace gradientand, possibly, hessian of f by their finite difference approximations, although thecomplexity of this operation is N + 1 computations of f for the gradient and evenhigher for the hessian. Whenever the computation of f has a critical complexity(and this is definitely the case when using minimization algorithms within a Semi-Lagrangian scheme), the minimization is better accomplished by cheaper techniques.3.3.3 Powell’s method and its modificationsThis method is genuinely derivative-free, but search directions are rather computedexploiting a geometric property of quadratic functions, which allows to constructconjugate directions without computing the gradient, nor the hessian. The basicidea is sketched in Figure 3.9 in the case of two space dimensions.Let a quadratic function f (shown through its elliptical level curves) and asearch direction d 1 be given. Consider the minima of f along two parallel lineswith direction d 1 (we recall that at a line-constrained minimum the search directionis tangent to the level curve). Then, the direction d 2 through the two minima isconjugate with d 1 . In Figure 3.9, this is recognized by the fact that d 2 passes throughthe center of the ellipses – this agrees with the fact that the second conjugatedirection brings to the global minimum, as it happens for a quadratic function intwo dimensions.

✐✐3.4. Numerical computation of the Legendre transform 73The technique can be extended to generate conjugate directions for a genericdimension N, and produces a scheme in the form (3.66) of a descent method. Being aconjugate direction method, Powell’s method converges in N iterations for quadraticfunctions of N variables, and is expected to be superlinear for smooth nonquadraticfunctions.Clearly, a certain number of technical details (which will not be reviewed here)should be set up to make the idea work in practice, but this method has inspired alarge number of efficient algorithms. Among the most successful related software,we mention the routine PRAXIS, which is freely available as an open-source Fortrancode.3.3.4 Trust-region methods based on quadratic modelsIn this class of schemes, the iterative minimization of the function f is carriedout by working at each step on an approximate (typically, quadratic) model of thefunction, which is associated to a so-called trust region, which is a set in which themodel is considered as a god approximation of the function.Let x k denote the current iterate. We write the quadratic model for thefunction f at x k asand the trust region as the ballm k (x k + s) = f(x k ) + (g k , s) + 1 2 (H ks, s), (3.69)B k = {x ∈ R N : ‖x − x k ‖ ≤ ∆ k }.Although the vector g k and the (symmetric) matrix H k play the role of respectivelythe gradient and the hessian of f, their construction should not use any informationon derivatives. In practice, they are typically constructed by interpolating a certainset of previous iterates where the function f has already been computed.Once the model (3.69) has been constructed at the iterate x k , a new candidatepoint x + kis computed as the minimum point for m k in the trust region. If the valuef(x + k ) represent a sufficient improvement of f(x k), then the iterate is updated asx k+1 = x + k, and typically a former point is removed from the set of point used inthe interpolation. If not, the trust region is reduced by decreasing ∆ k , and/or newpoints are added to improve the quality of the approximation.The robustness of the algorithm is crucially related to the geometric propertiesof the set of points used to construct the model (3.69), so various recipes have beenproposed to generate this set, to iteratively update it, and to improve it wheneverthe algorithm is unable to proceed. Once more, we will not present the technicaldetails, but rather point out that one such algorithm, NEWUOA, has available opensource implementations.3.4 Numerical computation of the LegendretransformIn the construction of SL schemes for Hamilton–Jacobi equations, a crucial roleis played by the Lax–Hopf formula, and therefore by the Legendre transform H ∗of the Hamiltonian function H. The possibility of an explicit computation of H ∗

✐✐74 Chapter 3. Elementary building blocksonly holds for a small class of functions, so we will address here the problem of itsnumerical approximation, focusing in particular on “fast” algorithms.We recall that the Legendre transform of the function H(p) is defined byH ∗ (q) = supp∈R d {p · q − H(p)}, (3.70)it takes finite values if the coercivity condition (2.26) is satisfied, and is a strictlyconvex function of q whenever H is a strictly convex function of p. Moreover, inthe specific framework of interest here, the sup is in fact a maximum, and can besearched for on a bounded set. In multiple dimensions, a “factorization formula”holds, so thatH ∗ (q) = supp 1∈R{{p 1 q 1 + supp 2∈Rp 2 q 2 + · · · + sup {p d q d − H(p)} · · ·p d ∈R}}.On the basis of this decomposition, it is possible to construct multidimensionalalgorithms for the Legendre transform by successive increases in the dimension.For this reason, we will sketch the basic algorithm in a single space dimension, thatis, using (3.70) with d = 1.Since the set in which we look for the max is bounded, it can be discretized bysetting up a grid of points p 1 , . . . , p N , with corresponding values H(p 1 ), . . . , H(p N )(note that in this case, the subscript refers to the index of a point, the problem beingone-dimensional). A parallel grid q 1 , . . . , q M (typically, with M ∼ N) is set up inthe variable q. The Discrete Legendre Transform algorithm consists in computing(3.70) at the discrete points q k , with the maximum obtained over the discrete set{p 1 , . . . , p N }, that isH ∗ (q k ) =max {p iq k − H(p i )} (k = 1, . . . , M). (3.71)i∈{1,...,N}In this form, the algorithm is convergent under natural assumptions, but has thedrawback of a quadratic complexity. This has prompted the search for faster implementations,and in particular with complexity O(N log N) or even O(N) as theversion outlined here.The ordered set of vertices (p i , H(p i )) defines a piecewise linear approximationof H(p) as shown in Figure 3.10, where the approximation between the nodes p iand p i+1 has the slopes i = H(p i+1) − H(p i ). (3.72)p i+1 − p iNote that, H being convex, this sequence is nondecreasing.Given q, the problem of computing H ∗ (q) is solved once known the argmax ofpq − H(p). On the other hand, a certain value p is in this argmax, if and only if q isincluded in the subdifferential of H at p. In the discretized version, depending onthe fact that p is a node or is internal to an interval of the grid, this subdifferentialmay be written as{D − [s i−1 , s i ] if p = p iH(p) =s i if p ∈ (p i , p i+1 )(this is shown in Figure 3.10) and consequently the argmax (as a function of q) hasthe form{p i if q ∈ [s i−1 , s i ]argmax {pq − H(p)} =(3.73)(p i , p i+1 ) if q = s i .

✐✐3.5. Commented references 75Figure 3.10. Computation of the Discrete Legendre Transform with piecewiselinear approximationIt suffices therefore to create the table of slopes (3.72) and to compare a given valueof q with the values of s i to find the argmax and hence H ∗ (q). Since a grid isalso set up in the q-domain, the comparison of the q k with the s i in (3.73) simplyrequires a merging of the two vectors. The final algorithm performs only operationsof linear complexity.3.5 Commented referencesMost of the numerical techniques reviewed in this chapter are analysed in detail indedicated monographs. Classical textbook on the approximation of Ordinary DifferentialEquations are [HNW93], [But03], but many others are available. The basictheory of polynomial interpolation is contained in any textbook in basic NumericalAnalysis, and any specific reference is useless. Finite element approximations are asomewhat more specialized (although well-established) issue, and among the manymonographs dedicated to this topic, we can quote [Cia02] and [BS08].On the contrary, to our knowledge no monograph has been devoted yet tonon-oscillatory schemes. A classical review on the basic ideas of ENO and WENOinterpolation is [Sh98] (from which we have reported most results concerning ENOinterpolation). WENO interpolation has been first studied in detail in [CFR05],proving in particular the results concerning positivity of the weights. Moreover, inaddition to what has been presented in this chapter, other reconstruction techniqueshave been applied to Semi-Lagrangian schemes. In particular, we mention here theso-called “shape preserving interpolations” (see [RW90] and the references therein),and, more recently, the Radial Basis Functions interpolations (see [Buh03] for ageneral review on RBF interpolation, and, for example, [Is03] for its application toSL schemes). The algorithm for tensorized multidimensional interpolations basedon a tree structure has been proposed in [CFF04], whereas a different approachto polynomial interpolation in high dimensions is provided by the so-called sparsegrids, for which an extensive review is given in [BG04].A general, up-to-date reference on optimization is [NW06]. Concerning directsearch methods, a more detailed review can be found in [KLT03], while a parallel(although less recent) review devoted to trust region methods is presented in

✐✐76 Chapter 3. Elementary building blocks[CST97]. Documentation and theory about two public domain optimization codes,PRAXIS and NEWUOA, can be found in respectively [Bre73] and [Pow08].Finally, the algorithm of Fast Legendre Transform (of complexity O(N log N))has been first proposed in [Br89], and further developed in [Co96]. More recently,the possibility of a linear-time (i.e., O(N)) algorithm has been shown in [Lu97].This algorithm corresponds to what has been presented in this chapter.

✐✐Chapter 4Convergence theoryThis chapter presents the main results of convergence for numerical schemes. In thecase of the Lax–Richtmeyer equivalence theorem for linear problems, the notions ofconsistency and stability are reviewed, with a special emphasis on monotone and L 2stability which play a crucial role for respectively the low-order and the high-orderschemes. In the case of convex HJ equations, we present the convergence results ofCrandall–Lions, Barles–Souganidis and Lin–Tadmor, which require ad hoc stabilityconcepts (monotonicity in the first two cases, uniform semiconcavity for the third).4.1 The general settingWe will carry out the discretization in the usual framework of difference schemes.Time is discretized with a (fixed) time step ∆t, so that t k = k∆t, whereas discretizationwith respect to space variables will require to set up a space grid in thecomputational domain (we make the standing assumption that this grid is nondegenerate,in a sense to be made explicit). We write a generic node as x j , j ∈ I, fora given set I of indices. Possible choices include:• unbounded structured uniform meshes in one space dimension, for which j ∈I = Z and x j = j∆x;• unbounded structured uniform meshes in multiple space dimensions, for whichthe node index j = (j 1 , . . . , j d ) is a multiindex, j ∈ I = Z d and x j =(j 1 ∆x 1 , . . . , j d ∆x d ) for a set of space discretization parameters ∆x 1 , . . . , ∆x d ≤∆x (these parameters are usually required to be linearly related one anotherto avoid degeneracy of the grid);• bounded structured uniform meshes in one or multiple space dimensions, forwhich I is the product of finite sets of indices, but nodes remain evenly spaced;• bounded unstructured meshes in one or more space dimensions (typicallybased on triangulations), for which I = {1, . . . , n n }, and each node is locatedwith its own coordinates. Suitable geometric conditions on trianglesensure in this case that the grid does not degenerate (see Chapter 3).Note that, in the unbounded case, it is usual to assume that the analyticalsolution be compactly supported. This makes it easier to develop a convergence77

✐✐78 Chapter 4. Convergence theorytheory in Hölder norms different from ‖ · ‖ ∞ , and will be implicitly done in thesequel.We denote by vjn be the desired approximation of u(x j , t n ), by V n and U(respectively, U(t)) the sets of nodal values for the numerical solution at time t n ,and for the exact solution u(x) (respectively, u(x, t)). W also denote by W and Φ(respectively, W (t) and Φ(t)) the sets of nodal values of generic functions w(x) andφ(x) (respectively, w(x, t) and φ(x, t)). In general, we will refer to the set of nodalvalues as to a (possibly infinite) vector.Convergence of numerical schemes will be analysed in a specific norm, andthroughout the book we will always use normalized Hölder norms, defined in thestructured uniform case by⎧⎨∑ ) 1/α(∆x‖W ‖ α := 1 · · · ∆x d j |w j| α if α < ∞(4.1)⎩max j |w j | if α = ∞.The natural matrix norm associated to the vector norm ‖ · ‖ α is defined as usual bywhich leads in particular, for α = 1, 2, ∞, to the forms∑‖B‖ 1 = sup |b ij |j∈Ii∈I(‖B‖ 2 = sup ρ m (B t B)‖BW ‖ α‖B‖ α := sup , (4.2)W ≠0 ‖W ‖ α‖B‖ ∞ = supi∈Im∈I∑|b ij |j∈I) 1/2(where ρ m (·) is a generic eigenvalue of the matrix inside the brackets). Note thatnatural matrix norms are insensitive to the scaling factor (∆x 1 · · · ∆x d ) 1/α appearingin (4.1).In the unstructured case, once written the numerical solution as a linear combinationof basis function in the form∑w j ψ j (x), (4.3)ja proper way of defining a Hölder norm is⎧⎛∣ ∫ ∣∣∣∣∣α ⎞1/α⎪⎨ ∑⎝ w‖W ‖ α :=j ψ j (x)dx⎠if α < ∞R d j ∣⎪⎩max j |w j | if α = ∞.(4.4)Once fixed the initial condition V 0 by setting v 0 j = u 0(x j ), the numerical schemewill be defined for n ≥ 0 by the iteration{V n+1 = S(∆; t n , V n )V 0 = U 0(4.5)

✐✐4.2. Convergence results for linear problems: the Lax–Richtmeyer theorem 79where ∆ = (∆x, ∆t) denotes the discretization parameters, with possible constraintsposed by stability and/or consistency requirements. If the dependence on the discretizationparameters needs not to be explicitly taken into account, we will simplywrite S(t n , V n ) instead of S(∆; t n , V n ); furthermore, if the evolution operator doesnot depend on t, we will write the scheme as S(∆; V n ), or in the simplified formS(V n ). Whenever useful, we will adopt the notation S ∗ (·) in which the asterisk, ifspecified, stands for a particular choice of the scheme. We will also denote by S j (·)the j–th component of S, that is, the scheme computed at x j .When dealing with linear problems, we will typically consider (and this isdefinitely the case when applying the equivalence theorem) schemes which are linearthemselves, that is, schemes in which the operator S(∆; t n , ·) is affine. Therefore,in the linear theory we examine schemes of the form{V n+1 = S(∆; t n , V n ) = B(∆; t n )V n + G n (∆)V 0 (4.6)= U 0 .In the same spirit as for the general case, the dependence on ∆ and/or t n willpossibly be omitted if redundant or unnecessary at all.4.2 Convergence results for linear problems: theLax–Richtmeyer theoremWe start the presentation of convergence theory for numerical methods with themore classical linear case, and in particular with time-invariant evolution operators.We assume therefore that the model to be approximated has the form{u t (x, t) + Au(x, t) = g(x, t), (x, t) ∈ Ω × [0, T ](4.7)u(x, 0) = u 0 (x)where A is a differential operator, and with suitable boundary conditions on ∂Ω, orwith Ω ≡ R d . Equation (4.7) will be assumed to be well-posed in some space H, tobe specified.Accordingly, the scheme will be assumed to be in the form{V n+1 = S(∆; V n ) = B(∆)V n + G n (∆)V 0 (4.8)= U 0 .It should be recalled from the very start that at the present state of the theory,the linear case is the only situation which allows for a general convergence theoryof numerical schemes. This theory is basically summarized in the Lax–Richtmeyerequivalence theorem.4.2.1 ConsistencyThe first requirement on a practical numerical scheme is that it should be a smallperturbation of the original equation. This idea, which is common to both the linearand the nonlinear case, is formalized in the concept of consistency.We define the local truncation error, or consistency error, asL(∆; t, U(t)) = 1 [U(t + ∆t) − S(∆; t, U(t))] (4.9)∆t

✐✐80 Chapter 4. Convergence theoryDefinition 4.1. Let U(t) be the set of nodal values of a solution u(x, t), and T > 0.The scheme S is said to be consistent if, for any t ∈ [0, T ],‖L(∆; t, U(t))‖ → 0 (4.10)as ∆ → 0, for any u 0 in a dense set of initial conditions.Remark 4.2. In the usual practice, at least in the linear case, the dense set ofsolutions on which (4.10) is tested coincides with C ∞ , which is dense in essentiallyall spaces of interest in the numerical approximation of PDEs. Moreover, oncechosen a smooth solution, the local truncation error typically satisfies a uniformbound on [0, T ], so that for a consistent schemeThis point will be implicitly assumed in the sequel.‖L(∆; t, U(t))‖ ≤ τ(∆) → 0. (4.11)Remark 4.3. The idea of consistency remains basically unchanged in the nonlinearcase, but its formal statement may vary somewhat in the various formulations, asit will be shown in section 4.4.4.2.2 StabilityThe second key assumption is that the small perturbations introduced by the discretizationshould not be amplified by the scheme to an uncontrolled magnitude.This leads to the definition of stability.Definition 4.4.n∆t ∈ [0, T ],The scheme (4.6) is said to be stable if, for any n such thatfor some constant M s > 0 independent of ∆.‖B(∆) n ‖ ≤ M s (4.12)Remark 4.5. The norm used in (4.12) must be an operator (matrix) norm compatiblewith the norm used in (4.10).Remark 4.6. While consistency is a standing assumption which also applies tononlinear situations, stability as stated by (4.12) may only be applied to linearschemes. In fact, some form of stablity is also necessary in nonlinear situations,but no such general concept exists, and each case is treated in a specific way.Assumption (4.12) is not easy to check in its greatest generality. Easier sufficientconditions for (4.12) to hold may be derived using the inequality (which holdsfor any matrix norm)‖B n ‖ ≤ ‖B‖ n .Then, a first trivial condition which implies stability is‖B‖ ≤ 1so that we would also have ‖B‖ n ≤ 1 for any n. This condition may be weakenedas‖B‖ ≤ 1 + C∆t (4.13)

✐✐4.2. Convergence results for linear problems: the Lax–Richtmeyer theorem 81for some C independent of the discretization parameters. In fact, taking into accountthat n ≤ T/∆t, we have‖B‖ n ≤ (1 + C∆t) n ≤≤ (1 + C∆t) T/∆t ≤≤ e CT .More in general, if ‖B‖ ∼ 1 + C∆t α , then the scheme is stable if α ≥ 1, unstable ifα < 1.4.2.3 ConvergenceThe definition of convergent scheme is related to the possibility of approximatingnot only a particular set of solutions (e.g., smooth solutions), but any solution inthe space H. So we will define a convergent scheme as follows.Definition 4.7.n∆t ∈ [0, T ],The scheme S is said to be convergent if, for any n such that‖V n − U(t n )‖ → 0 (4.14)for any initial condition u 0 ∈ H, as ∆ → 0.We have now all elements to state the main result, known as the Lax–Richtmeyerequivalence theorem:Theorem 4.8. Let the scheme (4.8) be consistent. Then, it is convergent if andonly if it is stable. Moreover, if the solution is smooth enough to satisfy (4.11),then, for any n such that n∆t ∈ [0, T ],for some positive constant C.‖V n − U(t n )‖ ≤ Cτ(∆) (4.15)Proof. We give a sketch of the proof, for the part related to sufficiency (stabilityimplies convergence), and in particular to (4.15). First, the scheme and theconsistency condition are rewritten at a generic step k ∈ [0, n − 1] as respectivelyV k+1 = B(∆)V k + G k (∆),U(t k+1 ) = B(∆)U(t k ) + G k (∆) + ∆tL(∆; t k , U(t k )).Subtracting now both sides, and defining the error at the step k ase k = U(t k ) − V k ,we havee k+1 = B(∆)e k + ∆tL(∆; t k , U(t k )).Taking now into account that e 0 = 0 by the choice of the initial condition, we obtain

✐✐82 Chapter 4. Convergence theoryat successive time steps:e 1 = ∆tL(∆; t 0 , U(t 0 ))e 2 = B(∆)e 1 + ∆tL(∆; t 1 , U(t 1 )) == ∆t (L(∆; t 1 , U(t 1 )) + B(∆)L(∆; t 0 , U(t 0 ))).e n = B(∆)e n−1 + ∆tL(∆; t n−1 , U(t n−1 )) == ∆t ( L(∆; t n−1 , U(t n−1 )) + · · · + B(∆) n−1 L(∆; t 0 , U(t 0 )) ) =n−1∑= ∆t B(∆) i L(∆; t n−i−1 , U(t n−i−1 ))i=0so that, due to the stability assumption, we can bound the error e n as:‖e n ‖ ≤ n∆tM s τ(∆) ≤ T M s τ(∆),and using also the assumption of consistency, we obtain that e n → 0, and in particular(4.15), once defined C = T M s .Remark 4.9. Note that the computation assumes that the consistency error wouldvanish, so that in principle it only holds for smooth solutions. On the other hand,smooth solutions are a dense set, so it is possible to prove convergence to all solutionsin H by a density argument.Remark 4.10. The reverse implication (convergence implies stability) requires theuse of the principle of uniform boundedness (Banach–Steinhaus theorem). We donot show this part of the proof, but remark that it allows to interpret the Courant–Friedrichs–Lewy condition, which will be described later, as a necessary conditionof stability.4.2.4 Time-dependent evolution operatorsWe turn now to the case of time-dependent evolution operators, that is{u t (x, t) + A(t)u(x, t) = g(x, t), (x, t) ∈ Ω × [0, T ]u(x, 0) = u 0 (x),(4.16)complemented again with suitable boundary conditions. Note that, in some respect,(4.16) might be considered as a model problem for a number of nonlinear models,in which linearization would lead to local evolution operators depending on thesolution, and therefore on time.A scheme for (4.16) should be assumed to be in the more general form (4.6),and adapting the Lax–Richtmeyer equivalence theorem to this case is quite straightforward.First, the general definition of consistency (4.10) directly applies to ascheme in the form (4.6). Second, retracing the proof of the equivalence theorem,the expression of the error at the n-th time step is replaced bye n = B(∆; t n−1 )e n−1 + ∆tL(∆; t n−1 , U(t n−1 )) == ∆t ( L(∆; t n−1 , U(t n−1 )) + B(∆; t n−1 )L(∆; t n−2 , U(t n−2 ) + · · · ++B(∆; t n−1 )B(∆; t n−2 ) · · · B(∆; t 1 )L(∆; t 0 , U(t 0 )) )

✐✐4.2. Convergence results for linear problems: the Lax–Richtmeyer theorem 83and it is immediate to see that the formulation (4.12) of stability should be replacedwith the requirement that, for any n such that n∆t ∈ [0, T ],‖B(∆; t n )B(∆; t n−1 ) · · · B(∆; t 1 )‖ ≤ M sfor some constant M s > 0 independent of ∆. In practice, in what follows we willnever apply the general stability condition (4.12), but rather the sufficient condition(4.13), which obviously works also in the time-dependent case once the constant Cis independent of the time step considered.4.2.5 The stationary caseTo conclude this review on linear convergence theory, we sketch the basic ideas totreat stationary problems. In particular, we consider here problems of the linearformAu(x) = g(x) (4.17)and, for our specific purposes, schemes in the fixed point formV = S(∆; V ) (4.18)in which a component v j of the vector V is intended to approximate u(x j ). In orderto ensure that the discrete problem has a unique solution, we typically assume thatS is a contraction, that is‖S(∆; V ) − S(∆; W )‖ ≤ L S < 1for V , W in some set U, such that S(∆; U) ⊆ U (note that in general the Lipschitzconstant L S should be assumed to depend on ∆). In this framework, the numericalsolution may be computed as a limit of the fixed point iteration{V (k+1) = S ( ∆; V (k))V (0) (4.19)∈ U.The similarity between (4.19) and (4.5) is more than formal. An iteration like(4.19) is often the result of applying a scheme in the time-dependent form (4.5) tocompute the stationary state (4.17) of the evolutive equation (4.7), provided such astationary state exists. In this strategy, which produces the so-called time-marchingschemes, the discrete solution is computed as a regime state of the scheme, just asthe exact solution is a regime state of the equation. Also, for all these reasons,when considering a time-marching scheme we might find it useful to introduce atime discretization parameter even if the equation to be approximated is stationary.As it has been remarked for the evolutive case, in the linear theory we considerschemes with S(·) affine, that is, schemes of the formV = S(∆; V ) = B(∆)V + G. (4.20)In this situation, the condition for S to be a contraction readsL S = ‖B(∆)‖ < 1 (4.21)and the invariant set U coincides with the whole domain of S. Note that, in theproof of convergence, (4.21) will appear as a stability condition. Note also that, ifthe fixed point system (4.20) is recast as a linear system in the form(I − B(∆))V = G,

✐✐84 Chapter 4. Convergence theoryand if condition (4.21) is enforced in the ∞-norm (which typically happens in monotoneschemes), then by (4.21) the matrix I − B(∆) is diagonally dominant, andtherefore nonsingular. Hence, the practical implementation of the scheme needsnot to be carried out in iterative form.ConsistencyWhile the general idea of plugging the exact solution in the scheme also applies toschemes in the form (4.18) or (4.20), a correct scaling should be adopted to definethe consistency error. We define the local truncation error for the scheme (4.18), asL(∆; U) =and the scheme is said to be consistent if11 − L S[U − S(∆; U)] , (4.22)‖L(∆; U)‖ ≤ τ(∆) → 0 (4.23)as ∆ → 0, for any u(x) in a dense set of solutions. When applying this definitionto the particular case of (4.20), the local truncation error is given byL(∆; U) =1[U − B(∆)U − G] . (4.24)1 − ‖B(∆)‖Remark 4.11. Typically, in time-marching schemes, the contraction coefficientsatisfies L S ∼ 1 − C∆t, and therefore the definition of local truncation error (4.22)matches the definition (4.9), when applied to a stationary solution.ConvergenceWe define convergence for the stationary scheme (4.18) in the obvious way, that is,requiring that‖V − U‖ → 0,as ∆ → 0. We can now give a convergence result which adapts the Lax–Richtmeyertheorem to the framework of linear stationary schemes in the form (4.20).Theorem 4.12. Let the scheme (4.20) be consistent and satisfy (4.21). Then, itis convergent. Moreover, if the solution is smooth enough to satisfy (4.23), then‖V − U‖ ≤ τ(∆). (4.25)Proof. Following the same guidelines of the equivalence theorem, the scheme andthe consistency condition are rewritten asV (k+1) = B(∆)V (k) + G,U = B(∆)U + G + (1 − ‖B(∆)‖)L(∆; U).Subtracting both sides, and defining the error at the iteration k ase (k) = V (k) − U,

✐✐4.3. More on stability 85we obtaine (k+1) = B(∆)e (k) + (1 − ‖B(∆)‖)L(∆; U).We are interested in giving a bound on the limitlimk→∞ e(k) = lim[U − V (k)]k→∞which corresponds to the error on the fixed point solution V of (4.20). Since thissolution is unique, it does not depend on V (0) and we can assume that V (0) = U,so that e (0) = 0, and we obtain for successive iterations:e (1) = (1 − ‖B(∆)‖)L(∆; U)e (2) = B(∆)e (1) + (1 − ‖B(∆)‖)L(∆; U) == (1 − ‖B(∆)‖)(I + B(∆))L(∆; U).e (k) = B(∆)e (k−1) + (1 − ‖B(∆)‖)L(∆; U) == (1 − ‖B(∆)‖) [ I + B(∆) + · · · + B(∆) k−1] L(∆; U).Now, giving an upper bound on the sum in square brackets as a geometric series,we can estimate the error e (k) as:‖e (k) 1‖ ≤ (1 − ‖B(∆)‖)‖L(∆; U)‖ ≤ τ(∆),1 − ‖B(∆)‖and, by consistency, we obtain that e (k) → 0, and in particular (4.25) if the solutionis smooth enough.4.3 More on stabilityWe turn back here to examine in greater detail the concept of stability, and inparticular some special forms of this general assumption. Most of what will be saidhere can be applied to linear as well as nonlinear problems and schemes.4.3.1 The CFL conditionIn 1928, long before the theory of convergence of numerical schemes for PDEs hadbecome an established matter, Courant, Friedrichs and Lewy published a paperin which the concept of domain of dependence was singled out as a key point forthe convergence of numerical schemes. To sketch the general idea, we first definethe analytical domain of dependence D d (x, t) of the solution u at a point (x, t), asthe smallest set such that, if the initial condition u 0 of (4.7) is changed on the setR N \ D d (x, t), u(x, t) remains unchanged. The discrete counterpart of this set isthe numerical domain of dependence D ∆ d (x, t) which, for a point (x, t) = (x i, t n ) inthe space-time grid, is the set of all nodes x j such that u 0 (x j ) affects the value v n i .Figure 4.1 illustrates these two concepts.The Courant, Friedrichs and Lewy (CFL) necessary condition states that, forany point (x, t), the analytical domain of dependence must be included in the limsupof numerical domains of dependence, that isD d (x, t) ⊆ Dd(x, 0 t) = lim sup Dd ∆ (x, t). (4.26)∆→0

✐✐86 Chapter 4. Convergence theoryFigure 4.1. Analytical and numerical domain of dependenceMore explicitly, the set Dd 0 (x, t) at the right-hand side of (4.26) is the set of all pointsfor which any open neighbourhood contains points of Dd ∆ (x, t) for some value of ∆,as ∆ → 0.The proof of the CFL condition is very simple: if (4.26) is not satisfied, thenthere exist subsets of D d (x, t) which have no intersection with Dd ∆ (x, t), at least as∆ → 0. Therefore, since changing the value of v 0 in these subsets changes the valueof v(x, t) but not the value of the numerical approximation, the scheme cannot beconvergent.On the other hand, by the equivalence theorem (which was stated about thirtyyears later) we can interpret this situation in more general terms. Since a consistent,convergent scheme is necessarily stable, if a consistent scheme violates the CFLcondition, then this scheme is unstable.4.3.2 Monotonicity and L ∞ stabilityFor a number of differential problems (and in particular for the model problemswhich have been considered in the previous chapter), two important qualitativeproperties are satisfied:• If u 0 is changed by adding a constant c, the corresponding solution is u(x, t)+c.In particular, in lack of source terms, constant solutions are preserved, thatis, if u 0 ≡ c, then u(x, t) ≡ c for any x and t > 0.• starting from two different initial conditions u 0 and w 0 such that u 0 (x) ≥w 0 (x), this ordering is preserved also for t > 0, that is, u(x, t) ≥ w(x, t);This makes it natural to require similar properties from the numerical scheme.In fact, we will see that this is a common, although very restrictive, notion ofstability in the nonlinear case.The first property is usually expressed in terms of invariance with respect tothe addition of constants, as follows:Definition 4.13. The scheme S is said to be invariant with respect to the additionof constants ifS(V + c) = S(V ) + c (4.27)for any vector V , where the sum of a vector with the constant c is to be intendedcomponent by component.

✐✐4.3. More on stability 87It is clear that, in order to preserve constants, (4.27) should be complementedwith the assumption that S(0) = 0.In particular, when the scheme is linear, the matrix B = (b ij ) must satisfyS i (V + c) = BV + Bc + G= S(V ) + c ∑ jb ijand therefore (4.27) is satisfied if and only if ∑ j b ij = 1 for any i (note that, hereand in the sequel, the dependence of the vector G on the time step is irrelevant).The concept of monotonicity requires some more words. We first define amonotone scheme:Definition 4.14. The scheme S is said to be monotone if, for any ∆,S(∆; V ) − S(∆; W ) ≥ 0 (4.28)for any couple of vectors V and W such that V − W ≥ 0, this inequality to beintended component by component.Usually, checking monotonicity of a scheme is relatively easy.scheme is linear, we may write for the i–th component:First, if theS i (V ) − S i (W ) = ∑ jb ij v j + g i − ∑ jb ij w j − g i == ∑ jb ij (v j − w j )so that monotonicity is satisfied if and only if all entries b ij of the matrix B arepositive. If a linear scheme is invariant for the addition of constants and monotone,then it is stable in l ∞ . In fact, under these assumptions the entries on a row of thematrix B must be positive and have unity sum, and hence the matrix has unity∞–norm.If the scheme is nonlinear, enforcing the conditionS i (V ) − S i (W ) ≥ 0for any V ≥ W requires that, for any i and j,∂∂v jS i (V ) ≥ 0.Note that any (linear or nonlinear) scheme which is monotone and preserves constantsis also L ∞ stable. In fact, defining two vectors W and W such that w j ≡sup x v 0 (x), w j ≡ inf x v 0 (x), we obtain thatW ≤ V 0 ≤ W , (4.29)and due to the two properties mentioned above, successive steps satisfyW = S n (W ) ≤ V n ≤ S n (W ) = W (4.30)

✐✐88 Chapter 4. Convergence theory(where S n denotes the n-iterated composition S ◦ S ◦ · · · ◦ S) so that the numericalsolution is uniformly bounded in L ∞ .More in general, it can be proved that if a scheme is invariant with respect tothe addition of constants and is monotone, then it is nonexpansive in L ∞ , that is‖S(V ) − S(W )‖ ∞ ≤ ‖V − W ‖ ∞ . (4.31)In fact, set the constant c = ‖(V − W ) + ‖ ∞ . Then, we also have that V ≤ W + c,so that monotonicity of the scheme implies thatand thereforeS(V ) ≤ S(W + c) = S(W ) + cS(V ) − S(W ) ≤ ‖(V − W ) + ‖ ∞which implies (4.31), once the roles of V and W are interchanged. Note that, inthe linear case, this is directly implied by the fact that B has unity ∞–norm.4.3.3 Monotonicity and Lipschitz stabilityA further property of monotone scheme, which is crucial in treating nonlinear problems,is nonexpansivity in the Lipschitz norm. This characteristic is not a trivialconsequence of monotonicity and conservation of constants, since it not only requiresthat L ∞ be conserved, but that Lipschitz constant be conserved too. Thisgap requires an additional assumption of invariance by translation which will beformally stated as follows.Definition 4.15. The scheme S is said to be invariant by translation if, definedthe translation operator Θ i such thatwe have, for any i = 1, . . . , d,(Θ i V ) j = V j+ei ,S(∆; Θ i V ) = Θ i S(∆; V ) (4.32)Clearly, this definition requires a uniform infinite space grid. Also, we do notexpect that it could be satisfied when treating variable coefficient equations. Now,define the i–th partial right incremental ratio at the point x j asD i,j [V ] = v j+e i− v j∆x i(i = 1, . . . , d). (4.33)Making use of the operator Θ iright–hand side of (4.33) aswe can write the maximum value over j of theIf this computation is performed on S(V ), we have‖D i [V ]‖ ∞ = ‖Θ iV − V ‖ ∞∆x i, (4.34)‖D i [S(V )]‖ ∞ = ‖Θ iS(V ) − S(V )‖ ∞∆x i= ‖S(Θ iV ) − S(V )‖ ∞∆x i, (4.35)

✐✐4.3. More on stability 89so that, applying the property of L ∞ –nonexpansivity of the scheme, we finallyobtain‖D i [S(V )]‖ ∞ ≤ ‖Θ iV − V ‖ ∞∆x i= ‖D i [V ]‖ ∞ . (4.36)Therefore, since none of the partial incremental ratios is expanded, this also holdsfor the Lipschitz constant.4.3.4 Von Neumann analysis and L 2 stabilityIt is clear that condition (4.12) is strictly related to the eigenvalues of B. Since anymatrix norm is bounded from below by the spectral radius, it would be natural torequire that, for a generic eigenvalue ρ m of B, the inequalityρ m (B) ≤ 1 + C∆t (4.37)should hold for a stable scheme. Condition (4.37) is usually referred to as VonNeumann stability condition and for the moment it will be considered a necessarycondition. Indeed, if (4.37) is not satisfied, then successive powers of B cannot beuniformly bounded, and the scheme is clearly unstable. Moreover, in general (4.37)is still far from being a sufficient condition, for the reasons we will soon explain.Let B be set in Jordan canonic form by a transformation T , so thatB(∆) = T (∆) −1 Λ(∆) T (∆) (4.38)(here, we have explicitly indicated that all these matrices depend on the discretizationparameters). Now, successive powers of B have the formand thereforeB(∆) n = T (∆) −1 Λ(∆) n T (∆) (4.39)‖B(∆) n ‖ ≤ ∥ ∥ T (∆)−1 ∥ ∥ · ‖Λ(∆) n ‖ · ‖T (∆)‖ . (4.40)Giving a bound on successive powers of B via (4.40) may not be easy, either. Threemain technical difficulties still hold:i) The transformation T itself depends on ∆, so that it is necessary to furthergive a uniform bound on the condition number K(T ) = ‖T −1 ‖ · ‖T ‖ as afunction of ∆;ii) If Λ has multiple eigenvalues, then the fundamental solutions associated tosuch eigenvalues are of the kind n ν ρ n . Even if |ρ| < 1, it is clear that thesesolutions are asymptotically stable, but may have extrema which are not easyto locate and bound;iii) The structure of eigenvectors is unknown in general, and this makes it difficultto look for eigenvalues.Points i) and ii) can be overcome by assuming B to be a normal matrix, thatis, a matrix which could be brought to diagonal form by means of an orthonormaltransformation T . Actually, if this is the case, then all eigenvalues are simple (andhence, the associated fundamental solutions are of the form ρ n ). Moreover, since Tis orthonormal, working in the 2–norm we also have, for any ∆,K 2 (T (∆)) = ∥ ∥ T (∆)−1 ∥ ∥2‖T (∆)‖ 2= 1.

✐✐90 Chapter 4. Convergence theoryTaking also into account that the 2–norm of a diagonal matrix is given by its spectralradius, we get therefore‖B(∆) n ‖ 2≤ K 2 (T (∆)) ‖Λ(∆) n ‖ 2= sup |ρ m (B(∆)) n | , (4.41)mand condition (4.37) becomes also sufficient.Concerning point iii), if one restricts to circulating matrices, which are a subclass ofnormal matrices, then the structure of eigenvectors is also known. More precisely, ifn n is the number of nodes and B ∈ R nn×nn is a circulating matrix, any eigenvectorV has components of the form v j = z j , with z a n n –th root of the unity. Writingmore explicitly such root of the unity in the form z = e imθ with θ = 2π/n n andm = 0, 1, . . . , n n − 1, we getv j = e i 2πmjnn , (4.42)and using this form in the relationshipρv j = ∑ lb jl v l , (4.43)which defines the eigenvalues, we obtainρ m e i 2πmjnn= ∑ lb jl e i 2πmlnn (4.44)that is, for any j,ρ m = ∑ lb jl e i 2πm(l−j)nn . (4.45)In the literature, the eigenvalue ρ m is also referred to as the amplification factorassociated to the harmonic component m. It depends on ∆ via the coefficients b jl .Assuming that B is a circulating matrix is really restrictive, but still correspondsto a real (in a sense, the simpler) situation of application of a given scheme.In particular, most schemes can be brought to this form when applied to constantcoefficient equations in one space dimension, with periodic boundary conditions andwith constant space discretization step. This is precisely the framework in whichVon Neumann condition (4.37) becomes sufficient for L 2 stability. Actually, thiscan be the only practical stability analysis for a number of schemes, and is widelyreputed to give at least an indication for more general cases.In general, we may not be really interested in computing the eigenvalue ρ mrelated to a specific harmonic component, but rather to prove the bound (4.37).Since (4.45) shows that there exists a mapping between the boundary of the unitydisk of C and a curve (always in C) containing the eigenvalues, it is possible toneglect the position of the specific eigenvalue on this curve, and rather focus on thefact that this curve itself be contained in the unity disc. On the other hand, if onekeeps the discretization steps constant and increases the number of nodes n n (thismeans enlarging the interval over which the equation is posed), then it becomesclear that the roots of the unity become denser on the original curve, and thereforethe eigenvalues become denser on the transformed curve. This makes it reasonableto treat the phaseω = 2πmn n∈ [0, 2π] (4.46)

✐✐4.4. Convergence results for Hamilton–Jacobi equations 91as a continuous parameter, and to carry out the stability analysis on the basis ofthe condition∣ ∑∣∣∣∣|ρ(ω)| =b jl (∆) e iω(l−j) ≤ 1 + C∆t. (4.47)∣lIn this form, the Von Neumann analysis is suitable to treat both the case of periodiccondition and the case posed on the whole of R (which is obtained as the limit casefor n n → ∞).Remark 4.16. Clearly, the eigenvalues of the matrix B(∆) could be estimatedby means of Gershgorin’s theorem, but condition (4.37), when ρ m (B) is estimatedwith respectively row or column sums, becomes equivalent to (4.13) written withrespectively the ∞– or the 1–norm of B.4.4 Convergence results for Hamilton–JacobiequationsWe move in this section to the study of the convergence theory for the convex HJequation,{u t (x, t) + H(Du(x, t)) = 0 (x, t) ∈ R d × [0, T ],(4.48)u(x, 0) = u 0 (x).Once out of the framework of linear equations (and linear schemes), no convergenceresult of the same generality of Lax–Richtmeyer theorem exists. In practice, whileconsistency of a scheme can be defined in essentially the same way for a nonlinearscheme, stability becomes a more subtle topic, since uniform boundedness ofnumerical solutions does not suffice in general for convergence.Therefore, different nonlinear stability concepts have been developed for nonlinearequations and schemes. In the framework of Hamilton–Jacobi equations twomain concepts have been proposed: the more classical idea of monotone stability(which leads to the convergence theorems of Crandall–Lions and Barles–Souganidis)and the idea of uniform semiconcavity used in the Lin–Tadmor convergence theorem.We also mention that, in the specific field of Semi-Lagrangian schemes, Lipschitzstability can be an equally useful concept.4.4.1 Crandall–Lions and Barles–Souganidis theoremsWe collect together these two results which make use of monotonicity as a stabilityassumption.The Crandall–Lions theorem is inspired by the result of convergence of monotoneconservative schemes for conservation laws, therefore it assumes the schemeto have a structure which parallels the structure of conservative schemes. On thecontrary, Barles–Souganidis theorem does not assume any particular structure forthe scheme, and is suitable for more general situation (including second-order HJequations), provided a comparison principle holds for the exact equation. It alsorequires a more technical definition of consistency.

✐✐92 Chapter 4. Convergence theorySchemes in differenced form: the Crandall–Lions theoremWe present this result referring to the case of two space dimensions, the extensionto an arbitrary number of dimensions being straightforward. With a small abuse ofnotation, we rewrite equation (4.48) asu t + H(u x1 , u x2 ) = 0. (4.49)Crandall–Lions theorem works in the framework of difference schemes, so we assumethat the space grid is orthogonal and uniform, ∆x i being the space step along thei–th direction. We define an approximation of the partial derivative u xi at the pointx j by the right (partial) incremental ratio (4.33), that isD i,j [V ] = v j+e i− v j∆x i(i = 1, 2).In parallel with the definition of schemes in conservative form for conservation laws,we define here the class of schemes in differenced form.Definition 4.17. A scheme S is said to be in differenced form if it has the formv n+1j = v n j − ∆tH (D 1,j−p [V n ], . . . , D 1,j+q [V n ]; D 2,j−p [V n ], . . . , D 2,j+q [V n ]) ,(4.50)for two multiindices p and q with positive components, and for a Lipschitz continuousfunction H (called the numerical Hamiltonian).In practice, (4.50) defines schemes in which the dependence on V n appears onlythrough its finite differences, computed on a rectangular stencil of points aroundx j . The differenced form of a scheme lends itself to an easier formulation of theconsistency condition, which is given in the followingDefinition 4.18. A scheme in differenced form is consistent if, for any a, b ∈ R,H(a, . . . , a; b, . . . , b) = H(a, b). (4.51)Remark 4.19. We point out that the previous definition matches the usual one.To show this fact, we rewrite the scheme in the formv n+1j− v n j∆t+ H (D 1,j−p [V n ], . . . , D 1,j+q [V n ]; D 2,j−p [V n ], . . . , D 2,j+q [V n ]) = 0.For the first term, it is clear that, once replaced V n by U(t n ),u(x j , t n+1 ) − u(x j , t n )∆t→ u t (x j , t n ),whereas for the second, it is necessary to notice that, for i = 1, 2 and for all fixedmultiindex k,D i,j+k [U(t n )] = u xi (x j , t n ) + O(∆x).Therefore, using this estimate in the scheme, we obtainH (D 1,j−p [U(t n )], . . . , D 1,j+q [U(t n )]; D 2,j−p [U(t n )], . . . , D 2,j+q [U(t n )]) =

✐✐4.4. Convergence results for Hamilton–Jacobi equations 93= H (u x1 (x j , t n ), . . . , u x1 (x j , t n ); u x2 (x j , t n ), . . . , u x2 (x j , t n )) + O(∆x),where all the error terms o(∆x) have been collected due to the Lipschitz continuityof H. On the other hand, applying (4.51), we obtain at the point (x j , t n ):H(u x1 , . . . , u x1 ; u x2 , . . . , u x2 ) + O(∆x) = H(u x1 , u x2 ) + O(∆x)which correspond to the usual notion of consistency.Also, in the nonlinear case we expect that monotonicity may or may not holddepending on the speed of propagation of the solution, this speed being related tothe Lipschitz constant of u 0 (in fact, in general monotonicity does depend on thespeed of propagation, but in the linear case this speed is given and unrelated tou 0 ). We will say that the scheme is monotone on [−R, R] if (4.14) is satisfied forany V and W such that |D i,j [V ]|, |D i,j [W ]| ≤ R.We have now all elements to state the Crandall–Lions convergence theorem.Theorem 4.20. Let H : R 2 → R be continuous, u 0 in (4.48) be bounded andLipschitz continuous (with Lipschitz constant L) on R 2 , and vj 0 = u 0(x j ). Let thescheme (4.50) be monotone on [−(L+1), L+1] and consistent, for a locally Lipschitzcontinuous numerical Hamiltonian H. Then, there exists a constant C such that,for any n ≤ T/∆t,∣ vnj − u(x j , t n ) ∣ ≤ C∆t1/2(4.52)for ∆t → 0, ∆x i = λ i ∆t, (i = 1, 2).Remark 4.21. We omit the proof, which is very technical. As a pure convergenceresult, Crandall–Lions theory is generalized by the Barles–Souganidis theorem, althoughin the latter result no convergence estimate is obtained.Remark 4.22. In the sequel, this convergence result will be applied to first-order,three-point stencil schemes in one space dimension. In this particular setting, adifferenced scheme takes the simpler form (in which the index i of the variable isclearly omitted)v n+1j = vj n − ∆tH (D j−1 [V n ], D j [V n ]) , (4.53)involving only the points v n j−1 , vn jand vn j+1in the computation of vn+1 j .Exploiting the comparison principle: the Barles–Souganidis theoremWhile it still requires monotonicity, Barles–Souganidis convergence theory [BaS91]gives a more abstract and general framework for convergence of schemes, includingthe possibility of treating second-order, degenerate and singular equations. Roughlyspeaking, this theory states that any monotone, stable and consistent scheme convergesto the exact solution provided there exists a comparison principle for thelimiting equation.The Cauchy problem under consideration is:{u t + H(x, u, Du, D 2 u) = 0 on R d × (0, T ),u = u 0 on R d × {t = 0}(4.54)

✐✐94 Chapter 4. Convergence theoryThe function H : R d ×R×R d ×M d → R (where M d is the space of d×d symmetricmatrices) is a continuous Hamiltonian, possibly depending on the Hessian matrixof second derivatives, and is assumed to be elliptic, i.e.H(x, w, p, A) ≤ H(x, w, p, B) (4.55)for all w ∈ R, x, p ∈ R d , A, B ∈ M d such that A ≥ B, that is, such that A − B isa positive semidefinite matrix. In particular, this form includes the first-order HJequation (4.48).We will assume that a comparison principle holds true for (4.54), i.e. we assume thatif u and v are respectively a super- and a sub-solution of (4.54) on R d × (0, T ) → R,and u(·, 0) ≤ v(·, 0), then u ≤ v.We consider a scheme in the general form (4.5). First, we require the property(4.27) of invariance with respect to the addition of constants. Then, a generalizedconsistency condition is assumed as follows.Definition 4.23. Let ∆ m ≡ (∆x m , ∆t m ) be a generic sequence of discretizationparameters, (x jm , t nm ) be a generic sequence of nodes in the space–time grid suchthat, for m → ∞,(∆x m , ∆t m ) → 0 and (x jm , t nm ) → (x, t). (4.56)Let φ ∈ C ∞ (R d × (0, T ]). Then, the scheme S is said to be consistent iflim infm→∞φ(x jm , t nm ) − S jm (∆ m ; Φ(t nm−1))∆t m≥ φ t (x, t) ++H(x, φ(x, t), Dφ(x, t), D 2 φ(x, t)),φ(x jm , t nm ) − S jm (∆ m ; Φ(t nm−1))lim sup≤ φ t (x, t) +m→∞∆t m+H(x, φ(x, t), Dφ(x, t), D 2 φ(x, t)).(4.57)Here, the index of the sequence is m, j m and n m denoting the correspondingindices of a node with respect to the m–th space-time grid, and we recall that byΦ or Φ(t) we denote the vector of node values for respectively φ(x) and φ(x, t).Moreover, H and H denote here lower and upper semicontinuous envelopes of H:H(x, φ(x, t), Dφ(x, t), D 2 φ(x, t)) =H(x, φ(x, t), Dφ(x, t), D 2 φ(x, t)) =lim inf H(y, φ(y, s), Dφ(y, s),(y,s)→(x,t) D2 φ(y, s)),lim sup H(y, φ(y, s), Dφ(y, s), D 2 φ(y, s)).(y,s)→(x,t)Note that, if H is continuous (and this is definitely the case for the first-order HJequation (4.48)), then in (4.57) the lim inf and the lim sup must coincide, and thedefinition reduces to the usual definition of consistency, unless for the fact that inprinciple φ needs not be a solution.The standard definition of monotonicity is also replaced by a generalized monotonicityassumption stated as follows.

✐✐4.4. Convergence results for Hamilton–Jacobi equations 95Definition 4.24. Let (∆x m , ∆t m ) and (x jm , t nm ) be generic sequences satisfying(4.56). Then, the scheme S is said to be monotone (in the generalized sense) if itsatisfies the following conditions:if v jm ≤ φ jm then S jm (∆ m ; V ) ≤ S jm (∆ m ; Φ) + o(∆t m ) (4.58)if φ jm ≤ v jm then S jm (∆ m ; Φ) ≤ S jm (∆ m ; V ) + o(∆t m ). (4.59)for any smooth function φ(x).Also in this case, we have that if a scheme is monotone in the usual form(4.14), then it also satisfies (4.58)–(4.59).Now, consider a numerical solution V n (with vjn and its piecewise constant (intime) interpolation v ∆t defined as:{v ∆t I[V n ](x) if t ∈ [t n , t n+1 ) ,(x, t) =v 0 (x) if t ∈ [0, ∆t).Here, I[V n ] is assumed to be a general interpolation operatorI[V n ](x) =∑ψ l (x)vl n , (4.60)x l ∈S(x)where {ψ l } is a basis of cardinal functions in R d (which in particular satisfy theproperty ∑ l ψ l(x) ≡ 1), and S(x) is the stencil of nodes involved for interpolatingat the point x. We assume for the moment that it is contained in a ball of radiusO(∆x) around x, and delay a detailed treatment of interpolation techniques toChapter 3.The interpolation operator has also to verify a relaxed monotonicity property:if v j ≤ φ j for any j such that x j ∈ S(x), then I[V ](x) ≤ I[Φ](x) + o(∆t) (4.61)if φ j ≤ v j for any j such that x j ∈ S(x), then I[Φ](x) ≤ I[V ](x) + o(∆t) (4.62)where V and φ denote vectors of node values of respectively a generic numericalsolution, and of a smooth function φ(x). Moreover, I[·] satisfies|I[Φ](x) − Φ(x)| = o(∆t). (4.63)Note that, once ∆t and ∆x are related one another, bounds (4.61)–(4.63) (whichare usually written in terms of the space discretization parameter) may also beunderstood in terms of ∆t.We can now state the extended version of the convergence result given in [BaS91]:Theorem 4.25. Assume (4.27), (4.57) and (4.58)–(4.63). Let u(x, t) be the uniqueviscosity solution of (4.54). Then v ∆t (x, t) → u(x, t) locally uniformly on R d ×[0, T ]as ∆ → 0.Proof. Let the bounded functions u, u be defined byu(x, t) =lim sup(y,s)→(x,t)∆t→0v ∆t (y, s), u(x, t) = lim inf(y,s)→(x,t)∆t→0v ∆t (y, s). (4.64)

✐✐96 Chapter 4. Convergence theoryWe claim that u(x, t), u(x, t) are respectively a sub- and a super-solution of (4.54).Assume for the moment that the claim is true; then by the comparison principleu(x, t) ≤ u(x, t) on R d × (0, T ]. Since the opposite inequality is obvious by thedefinition of u(x, t) and u(x, t), we haveu = u = uand u is the unique continuous viscosity solution of (4.54). This fact together with(4.64) also imply the locally uniform convergence of v ∆t to u.Let us prove the previous claim. Let (x, t) be a local maximum of u − φ onR d × (0, T ] for some φ ∈ C ∞ (R d × (0, T ]). Without any loss of generality, we mayassume that (x, t) is a strict global maximum for u − φ and that u(x, t) = φ(x, t).Then, by a standard result from viscosity theory, there exist two sequences ∆t m ∈R + and (y m , τ m ) ∈ R d × [0, T ], which are global maximum points for v ∆tm − φ, andas m → ∞:∆t m → 0, (y m , τ m ) → (x, t), v ∆tm (y m , τ m ) → u(x, t).Then, for any x and t we havev ∆tm (x, t) ≤ φ(x, t) + ξ m (4.65)with ξ m = ( v ∆tm − φ ) (y m , τ m ) (note that u(x, t) = φ(x, t), and hence ξ m → 0).Since, in general, (y m , τ m ) is not a grid point, we need to reconstruct the valueattained by v ∆tm at such points. By the definition of v ∆t , there exist a t nm suchthat τ m ∈ [t nm , t nm+1) and v ∆tm (y m , τ m ) = v ∆tm (y m , t nm ). Furthermore, by thedefinition of I[·] in (4.60), there exist a set of nodes S(y m ) such thatI [V nm ] (y m ) =∑x j∈S(y m)ψ j (y m )v nmj . (4.66)Next, we apply (4.65) at t = t nm−1, x = x j ∈ S(y m ) and deduce, from (4.27) andthe monotonicity property (4.58), thatS j(∆m ; V nm−1) ≤ S j (∆ m ; Φ(t nm−1)) + ξ m + o(∆t m ).Recalling that the left-hand side is nothing but v nmj , we havewhich yelds, applying (4.60), (4.61):v ∆tm (y m , τ m ) ≤v nmj ≤ S j (∆ m ; Φ(t nm−1)) + ξ m + o(∆t m ),∑x j∈S(y m)ψ j (y m )S j (∆ m ; Φ(t nm−1)) + ξ m + o(∆t m ).Now, by the definition of ξ m , we getφ(y m , τ m ) ≤∑ψ j (y m )S j (∆ m ; Φ(t nm−1)) + o(∆t m ) (4.67)x j∈S(y m)We claim now that φ(y m , τ m ) = φ(y m , t nm )+O(∆t 2 m). In fact, either τ m = t nm(and the claim obviously holds), or τ m ∈ (t nm−1, t nm ). In the latter case, since

✐✐4.4. Convergence results for Hamilton–Jacobi equations 97(v ∆tm − φ)(y m , ·) has a maximum in τ m and v ∆tm is constant in (t nm−1, t nm ), thenφ t (y m , τ m ) = 0 and we have φ(y m , τ m ) = φ(y m , t nm ) + O(∆t 2 m).Using the previous claim in (4.67), we haveφ(y m , t nm ) ≤∑ψ j (y m )S j (∆ m ; Φ(t nm−1)) + o(∆t m ) (4.68)and, by (4.63),x j∈S(y m)φ(y m , t nm ) = I[Φ(t nm )](y m ) + o(∆t m ) =Now, (4.68) and (4.69) implylim infm→∞∑x j∈S(y m)∑x j∈S(y m)ψ j (y m )φ(x j , t nm ) + o(∆t m ).ψ j (y m ) φ(x j, t nm ) − S j (∆ m ; Φ(t nm−1))∆t m+ o(1) ≤ 0.Finally, by the consistency property (4.57), we obtain the desired result:φ t (x, t) + H(x, φ(x, t), Dφ(x, t), D 2 φ(x, t)) ≤ 0.(4.69)The proof that u is a super-solution follows the same arguments, unless for replacing(4.58) with (4.59). We leave this adaptation to the reader.Note that the consistency condition (4.57) might be reformulated so as to avoidany dependence on the variable t. In fact, adding and subtracting φ(x j , t n−1 ), weobtainφ(x j , t n ) − S j (Φ(t n−1 ))= φ(x j, t n ) − φ(x j , t n−1 )+ φ(x j, t n−1 ) − S j (Φ(t n−1 )).∆t∆t∆t(4.70)In the right–hand side of (4.70), the first term necessarily converges to φ t (x j , t n ),so that (4.57) is equivalent toH(x, φ(x), Dφ(x), D 2 φ(x)) ≤ lim infm→∞≤ lim supm→∞φ(x jm ) − S jm (∆ m ; Φ)≤ (4.71)∆t mφ(x jm ) − S jm (∆ m ; Φ)∆t m≤ H(x, φ(x), Dφ(x), D 2 φ(x)),for a function φ(x) depending on x alone, and x jm → x.Once in the form (4.71), the definition of consistency error parallels the correspondingdefinition (4.22) for time-marching schemes. In fact, Barles–Souganidistheorem can also be recast to work for schemes in the time-marching form (4.18)–(4.19), as follows.Theorem 4.26. Let u(x) be the unique viscosity solution in R d of the equationH(x, u, Du, D 2 u) = 0.Assume that the scheme is in the form (4.18), with S(∆, ·) a contraction. Assumemoreover that (4.27), (4.71), (4.58)–(4.63) hold, and that L S = 1 − C∆t + o(∆t).Then, I[V ](x) → u(x) locally uniformly on R d as ∆ → 0.

✐✐98 Chapter 4. Convergence theory4.4.2 Lin–Tadmor theorem and semi–concave stabilityIn Lin–Tadmor convergence theory, a different concept of stability is singled out.Once more, this appears to be an adaptation to HJ equations of a parallel resultobtained for conservation laws: as Crandall–Lions theorem derives from the classicalconvergence theory of monotone conservative schemes, Lin–Tadmor theorem derivesfrom the Lip ′ –stability theory.We start by giving the main concept of semi-concave stability:Definition 4.27. A family of approximate solutions u ɛ of (4.48) is said to besemi-concave stable if there exists a function k(t) ∈ L 1 ([0, T ]) such thatfor t ∈ [0, T ].D 2 u ɛ (x, t) ≤ k(t)I (4.72)More explicitly, condition (4.72) means that D 2 u ɛ (x, t) − k(t)I is negativesemidefinite, that is (since we are dealing with symmetric matrices), that all eigenvaluesof D 2 u are bounded from above by k(t). The core of the theory is a firstabstract result of convergence for perturbed semi-concave solutions.Theorem 4.28. Consider problem (4.48) for a semi-concave initial condition u 0with compact support, and assume the family u ɛ is semi-concave stable. Define thetruncation error associated to u ɛ asThen, for any t ∈ [0, T ],F (x, t) = u ɛ t(x, t) + H(Du ɛ (x, t)). (4.73)‖u(t) − u ɛ (t)‖ L 1 (R d ) ≤ C 1 ‖u 0 − u ɛ (0)‖ L 1 (R d ) + C 2 ‖F ‖ L 1 (R d ×[0,T ]). (4.74)Proof. Let the error be defined bye(x, t) = u(x, t) − u ɛ (x, t).By (4.48) and (4.73), the error satisfies the advection equation{e t (x, t) + G(x, t) · De(x, t) = F (x, t),e(x, 0) = u 0 (x) − u ɛ (x, 0)(4.75)where the advecting speed G is defined as the averageG(x, t) =∫ 10∂H (ηDu(x, t) + (1 − η)Du ɛ (x, t) ) dη. (4.76)∂pIn order to estimate the norm ‖e(T )‖ of the error at a given time T , we considernow the dual equation,{ψ t (x, t) + div ( G(x, t)ψ(x, t) ) = 0,(4.77)ψ(x, T ) = ψ T (x),

✐✐4.4. Convergence results for Hamilton–Jacobi equations 99where ψ(T ) is smooth and has support in a compact set Ω(T ). From (4.75) and(4.77), using Green’s formula, the L 2 scalar product ( e(t), ψ(t) ) satisfies∫d ( )e(t), ψ(t) + e(x, t)ψ(x, t)G(x, t) · n ds = ( F (t), ψ(t) )dt∂Ω(t)with Ω(t) a set depending on t, including the support of ψ(x, t) at T = t. Hence,the boundary integral vanishes at T , and we obtain∫( ) ( ) T( )e(T ), ψT = e(0), ψ(0) + F (t), ψ(t) dt ≤0≤ ‖e(0)‖ L 1 (R d )‖ψ(0)‖ L ∞ (R d ) + ‖F ‖ L 1 (R d ×[0,T ])which implies, via Hölder’s inequality,‖ψ(0)‖ L∞ (R‖e(T )‖ L1 (R d ) ≤ supd )‖e(0)‖ψ T‖ψ T ‖L1 (R d ) +L∞ (R d )supt∈[0,T ]‖ψ(t)‖ L ∞ (R d ),sup t∈[0,T ] ‖ψ(t)‖ L∞ (R+ supd )‖F ‖ψ T‖ψ T ‖L1 (R d ×[0,T ]), (4.78)L∞ (R d )and hence (4.74), as soon as we are able to bound on ‖ψ(t)‖ ∞ over the interval[0, T ], in terms of the final condition ψ T .To this end, we consider again (4.77), which givesddt ‖ψ(t)‖ L ∞ (R d ) + ( div G(x 0 , t) ) ‖ψ(t)‖ L∞ (R d ) = 0. (4.79)To derive (4.79), we multiply (4.77) by the sign of ψ(x, t) and denote by x 0 themaximum point of |ψ| at the time t. By Gronwall’s inequality, we obtain from(4.79), for t ∈ [0, T ]:( ∫ )T( )‖ψ(t)‖ L ∞ (R d ) ≤ ‖ψ T ‖ L ∞ (R d ) exp sup div G(x, s) ds , (4.80)xwhere by the definition (4.76), the divergence of G reads⎛⎞∫ 1div G = ⎝η ∑ ∂ 2 H ∂ 2 u+ (1 − η) ∑ ∂ 2 H ∂ 2 u ɛ⎠ dη. (4.81)0 ∂pi,j i ∂p j ∂x i ∂x j ∂pi,j i ∂p j ∂x i ∂x j(note that, in (4.81), arguments have been omitted since we are looking for a uniformbound). The first sum in (4.81) may be estimated as∑ ∂ 2 H ∂ 2 u= trace (D 2 H D 2 u) =∂p i ∂p j ∂x i ∂x ji,jt= trace ( (D 2 H) 1/2 D 2 u (D 2 H) 1/2) ≤≤ d · k(s) · β.Here, we have used the facts that eigenvalues of D 2 u are bounded from above byk(t), and eigenvalues of D 2 H are bounded from above by β, and that the trace is

✐✐100 Chapter 4. Convergence theorythe sum of eigenvalues. Since (unless for replacing u with u ɛ ) a similar estimateholds for the second sum in (4.81), we finally obtain()‖ψ(t)‖ L ∞ (R d ) ≤ ‖ψ T ‖ L ∞ (R d ) expwhich proves (4.78), and therefore (4.74).dβ∫ T0k(s)dsIn practice, this result needs some refinement in order to be applied to a numericalscheme. In particular, the concept of semi-concave stability must be weakenedand the measure of the local truncation error should be made more explicit.Discrete semi-concave stability The assumption of semi-concave stability, as statedby (4.72), is too strict for numerical solutions produced by a real scheme. In fact,a reconstruction of the solution, typically in piecewise polynomial form, may notsatisfy the upper bound on second derivatives at the interface between two cells ofthe grid, even in a monotone scheme. To overcome this problem, a discrete counterpartof the semi-concave stability is defined for the numerical solution v ∆ by abound on the second directional incremental ratios in the form:v ∆ (x + δ, t) − 2v ∆ (x, t) + v ∆ (x − δ, t)‖δ‖ 2 ≤ k(t). (4.82)Here, the function k(t) ∈ L 1 ([0, T ]) plays the same role as in the original definition,and δ is a vector whose norm should remain bounded away from zero, more precisely‖δ‖ ≥ C∆x. (4.83)It is possible to prove (see [LT01]) that (4.82)–(4.83) imply that each numericalsolution is close to a function of a family satisfying (4.72), so that theorem 4.28may be applied.Computation of the truncation error The second ingredient of the Lin–Tadmorconvergence theorem, the L 1 estimation of the truncation error, may also be difficultin general. In particular, (4.73) shows a somewhat reversed approach in measuringthe truncation error, that is, by plugging the numerical solution into the exactequation.An easier expression can be derived for Godunov-type schemes (as Semi-Lagrangian schemes are), taking into account that in this case numerical errorsare generated only by the projection step, and not by the evolution operator, whichis in principle exact. This expression will be introduced later, in the analysis of SLschemes for HJ equations.4.5 Numerical diffusion and dispersionConvergence analysis is concerned with the asymptotic behaviour of a scheme. Inparticular, its endpoint is the possibility of generating numerical solutions as closeas necessary to the exact one, although, in practice, we might be more interestedin examining the qualitative response of the scheme for a finite, practical value ofthe discretization parameters. In a different and complementary analysis, we regard

✐✐4.5. Numerical diffusion and dispersion 101Figure 4.2. Advection by a diffusive (left) or dispersive (right) scheme.the values vjn of the numerical solutions as samples of a function v(x, t) defined onR × [0, T ], so that vjn = v(x j, t n ). The function v is characterized in turn as thesolution of a Partial Differential Equation obtained by a perturbation of the exactPDE. This perturbed equation is known as modified equation.Typically, such an analysis is performed on the constant coefficient advectionequation, and as a result the modified equation usually takes the formv t + cv x = ν ∂kv + o(ν). (4.84)∂xk The coefficient ν = ν(∆) depends on the discretization parameters and, for convergencereasons, one expects that ν(∆) → 0 with the same order of the consistencyerror. Note that, in (4.84), we have made explicit the relevant perturbation, whereasthe o(ν(∆)) accounts for all the higher-order terms. In fact, we do not expect ingeneral that a finite number of perturbing terms could suffice to completely characterizethe function v.For k = 2 the modified equation appears as an advection–diffusion equation,and in this case we term the behaviour of the scheme as diffusive, and the coefficientν as the numerical viscosity (a positive number in stable schemes). If k > 2,the scheme is termed as dispersive. We show in Fig. 4.2 the typical response ofrespectively a diffusive and a dispersive scheme on the advection of a characteristicfunction. In the first case, since the fundamental solution is represented by theadvection of a gaussian kernel (which is positive), the modified equation satisfies amaximum principle – this behaviour is in fact characteristic of monotone schemes.In the second case, the fundamental solution is no longer positive and this resultsin the occurrence of under- and overshoots.Rather then outlining a general procedure for obtaining the modified equation,we will delay its practical construction to future examples. In difference schemes,the usual technique is to express the numerical samples vjn as Taylor expansionson v, but this technique is unsuitable for Semi-Lagrangian schemes. To performa diffusion/dispersion analysis on SL schemes, we derive therefore an analogous ofthe representation formula (1.3) for equation (4.84). Retracing the proof of (1.3),we can writed∂k[v(ξ + ct, t)] = ν v(ξ + ct, t) + o(ν)dt ∂xk

✐✐102 Chapter 4. Convergence theoryso that integrating between t n and t n+1 we obtain∫ tn+1∂ kv(x j , t n+1 ) = v(x j − c∆t, t n ) + νt n∂x k v(x j − c(t n+1 − t), t)dt + o(ν∆t),and since, by a zeroth order approximation,∫ tn+1∂ kνt n∂x k v(x j − c(t n+1 − t), t)dt = ν∆t ∂k∂x k v(x j − c∆t, t n ) + O(ν∆t 2 ),we finally getv(x j , t n+1 ) = v(x j − c∆t, t n ) + ν∆t ∂k∂x k v(x j − c∆t, t n ) + o(ν∆t). (4.85)In practice, in the operation of a SL scheme, the term associated to the k–th spacederivative of v is generated by the interpolation error, and this makes it possible torecover all terms of the modified equation, as it will be seen later on.4.6 Commented referencesThe Lax–Richtmeyer equivalence theorem has been first published in [LR56], thuscompleting a theoretical study which had started from the work of Courant, Friedrichsand Lewy (written in 1928, reprinted and translated into English in 1967 [CFL67]).By that time, the concept of stability had already been extensively studied andsingled out as a crucial requirement for convergence of schemes, along with variousstability criteria, including Von Neumann condition [VR50]. Good reviews onthe classical theory of convergence for difference schemes (including the study ofmodified equations) have been given in [RM67], and, more recently, in [Str89].In the field of HJ equations, convergence results for monotone schemes havebeen formulated at the very start of the theory of viscosity solutions. In additionto the convergence theorem, Crandall–Lions paper [CL84] also adapts to differenceschemes the results of L ∞ -nonexpansivity and of Lipschitz stability for monotoneschemes which had been first proved in a simple and abstact way in [CT80].Barles–Souganidis theorem appears in its first version in [Sou85a], but has beenrefined in [BaS91] to include second-order equations, like the equation of MeanCurvature Motion. We have chosen to present a version of the theorem more suitablefor real schemes [CFF10], since the original has a more abstract formulation. Finally,Lin–Tadmor convergence theory has been proposed in [LT01], and is inspired by asimilar result obtained for conservation laws [T91].

✐✐Chapter 5First-order approximationschemesThis chapter is devoted to a comparison of the basic SL scheme (proposed byCourant, Isaacson and Rees in 1952) with upwind and centered (Lax–Friedrichs)schemes. For each of the schemes, the presentation includes• Construction of the scheme• Consistency• Stability (CFL condition, monotonicity, Von Neumann analysis, uniform semiconcavity)• Convergence estimates• Numerical viscosityA comparison among the schemes is made, and simple numerical examples inone space dimension are presented to support our theoretical study. Note that, forthe sake of clarity, most of the presentation is made on the model problems and inone space dimension, but the way to treat more general cases (including multipledimensions) is sketched at the end of the sections, and further extensions of interestfor applications will be considered in the following chapters. A subsection is alsodevoted to the treatment of boundary conditions.5.1 Treating the advection equationIn this section, we introduce the schemes on the simpler case of the advectionequation. We will start with the basic case of the constant coefficient case,{u t (x, t) + cu x (x, t) = 0 (x, t) ∈ R × [0, T ], c > 0(5.1)u(x, 0) = u 0 (x)and generalize the ideas to the case with variable coefficients,{u t (x, t) + f(x, t)u x (x, t) = g(x, t) (x, t) ∈ R × [0, T ]u(x, 0) = u 0 (x).(5.2)Note that the simplified model (5.1) is also useful to perform L 2 (Von Neumann)stability analysis, as well as the study of numerical viscosity.103

✐✐104 Chapter 5. First-order approximation schemes5.1.1 Upwind discretization: the First Order Upwind schemeAs we have seen, in the physical and mathematical modeling of advection the advectionspeed (c or f(x, t)) is recognized as the vectorfield which makes the solutionpropagate. A first, natural strategy is therefore to let the numerical scheme mimicthis behaviour. More explicitly, upwind schemes are geometrically constructed sothat the numerical domain of dependence follows the direction of propagation ofthe solution.Construction of the schemeAlthough the upwind scheme could be derived in a different framework, like the fullydiscrete structure of conservative schemes, we will sketch its construction here byusing the method of lines, that is, introducing an intermediate semi-discrete version,which in our case will be understood as a formulation in which space derivatives(but not time derivatives) have been replaced by finite differences. In particular,we first focus on the constant coefficient case (5.1), then turn to the more generalcase (5.2).The constant coefficient equation Starting with the constant-coefficient equation(5.1), and approximating the space derivative u x (x j , t) with the incremental ratiobetween x j−1 and x j , the semi-discrete scheme computed at x j reads˙v j (t) + c v j(t) − v j−1 (t)∆x= 0. (5.3)A further discretization in time of (5.3) by means of the forward Euler method givesv n+1j− v n j∆t+ c vn j − vn j−1∆x= 0, (5.4)that is,v n+1j = vj n − c∆t [vn∆x j − vj−1n ]. (5.5)Setting λ = c∆t/∆x, the scheme also takes the more explicit formv n+1j = λv n j−1 + (1 − λ)v n j . (5.6)A key point in the first-order upwind scheme (which also causes the presence ofthe term upwind) is that the incremental ratio used to approximate u x should beperformed on the correct side of x j . While the importance of this point will bediscussed in relationship with stability, we note that in the previous discussion, thechoice of a left incremental ratio is related to the positive sign of c. A negative signwould rather require to use the right incremental ratio.The variable coefficient equation When applying the upwind strategy to equation(5.2), we must therefore discriminate on the basis of the sign of f(x, t): wheneverf(x j , t n ) > 0, the space derivative is replaced by a left incremental ratio, andvice versa if f(x j , t n ) < 0 (in case f(x j , t n ) = 0, the equation simply says thatu t (x j , t n ) = g(x j , t n )). It is convenient to consider the upwind scheme in the form(5.4), which immediately gives, for f(x j , t n ) > 0,v n+1j− v n j∆t+ f(x j , t n ) vn j − vn j−1∆x= g(x j , t n ), (5.7)

✐✐5.1. Treating the advection equation 105and for f(x j , t n ) < 0,v n+1j− v n j∆t+ f(x j , t n ) vn j+1 − vn j∆x= g(x j , t n ). (5.8)To sum up, the form of the scheme for f(x, t) changing sign is⎧vj n − f(x j , t n ) ∆t [vn∆x j − v n ]j−1 + ∆tg(xj , t n )⎪⎨if f(x j , t n ) > 0v n+1j = vj n + ∆tg(x j, t n ) if f(x j , t n ) = 0Consistency⎪⎩ vj n − f(x j , t n ) ∆t∆x[vnj+1 − vjn ]+ ∆tg(xj , t n ) if f(x j , t n ) < 0.(5.9)In order to analyse consistency, we apply the upwind scheme in the form (5.9) toa solution. For example, if f(x j , t n ) > 0, we estimate a single component of theconsistency error (4.9) asL Upj (∆; t, U(t)) = 1 [u(x j , t + ∆t) − u(x j , t) +∆t+f(x j , t) ∆t]∆x (u(x j, t) − u(x j−1 , t)) − ∆tg(x j , t) == u(x j, t + ∆t) − u(x j , t)∆t−g(x j , t) =+ f(x j , t) u(x j, t) − u(x j−1 , t)∆x−= u t (x j , t) + O(∆t) + f(x j , t)u x (x j , t) + O(∆x) − g(x j , t) == O(∆t) + O(∆x)where the two last displays use the fact that the error in discretizing the spaceand time derivatives is of first order if the solution is smooth (or at least, if ithas bounded second derivatives in space and time), and that (5.2) is satisfied by asolution. Since an analogous computation can be carried out with f(x j , t n ) > 0, wecan conclude that ∣ ∣∣L Upj (∆; t, U(t)) ∣ ≤ C(∆t + ∆x)and therefore, passing to a generic normalized Hölder norm (in particular, the norms‖ · ‖ 2 and ‖ · ‖ ∞ ), we can state the consistency of the scheme by∥ L Up (∆; t, U(t)) ∥ ∥ ≤ C(∆t + ∆x). (5.10)Remark 5.1. Consistency of the upwind scheme would not require any particularrelationship between the discretization steps, although a balance of the two termssuggests using steps of the same order of magnitude. In practice, this choice is alsorequired for stability, as we will soon show.

✐✐106 Chapter 5. First-order approximation schemesFigure 5.1. Numerical domain of dependence for the upwind scheme.StabilityIn the upwind scheme, all stability criteria lead to the same condition on the discretizationsteps, at least in the linear case. As we will see, this is no longer true innonlinear situations.CFL condition For simplicity, let us refer to the constant coefficient case with c >0, that is, to the scheme in the version (5.6). The numerical solution vj n at (x j, t n )depends on v n−1j−1 and vn−1 j , and in practice, stepping back in time, we construct anumerical domain of dependence in the form of a triangle, which is shown in Figure5.1 along with the characteristic passing through (x j , t n ). Due to the choice ofapproximating space derivatives with left finite differences, the triangle stands onthe left of x j – this agrees with the fact that, if c > 0, the characteristic passingthrough (x j , t n ) is located precisely on this side. To fulfill the CFL condition, wemust further impose that the foot of the characteristic x j − cn∆t (which representsthe analytical domain of dependence D d (x j , t n )) should be included in the numericaldomain of dependence [x j − n∆x, x j ]. This happens if cn∆t < n∆x, or moreexplicitlyλ = c∆t ≤ 1. (5.11)∆xClearly, the constant slope 1/c of characteristics in the x–t plane should be replaced,in the case of variable coefficients, by their minimum (positive or negative) slope±1/‖f‖ ∞ , so that in this case the CFL condition takes the form‖f‖ ∞ ∆t∆x(note that ‖ · ‖ ∞ stands here for the norm in L ∞ (R × [0, T ])).≤ 1 (5.12)Monotonicity We recall that the general condition for the monotonicity of a linearscheme in the form V n+1 = B(∆; t n )V n +G n (∆) (see Chapter 4) is that the entriesof the matrix B(∆; t n ) should be positive for any n. In the case of the upwind

✐✐5.1. Treating the advection equation 107scheme, and assuming that f(x j , t n ) > 0, the only nonzero entries on the j–th rowareb Upj,j−1 (∆; t n) = f(x j , t n ) ∆t∆xb Upj,j (∆; t n) = 1 − f(x j , t n ) ∆t∆x(note that they have unity sum so that the scheme is also invariant with respect tothe addition of constants). Since the first one is necessarily positive, the scheme ismonotone provided 1 − f(x j , t n ) ∆t∆x> 0 for any j and n, this being satisfied if‖f‖ ∞ ∆t∆x< 1. (5.13)Repeating the computations at points where f(x j , t n ) < 0, we obtain that monotonicityis ensured under the same condition.Von Neumann analysis We refer here to the scheme in the form (5.6). Given aneigenvector V of the scheme, in the form v j = e ijω , the condition which characterizesρ as an eigenvalue readsρv j = λv j−1 + (1 − λ)v j ,that is,ρ = (1 − λ) + λ cos ω + iλ sin ωSumming up, eigenvalues are located on a circle of the complex plane centered at1 − λ and with radius λ (note that this is precisely the boundary of the Gershgorindiscs of the matrix B Up ). Therefore, it results that for λ ∈ [0, 1] the eigenvalues arecontained in the unity disc, and the Von Neumann condition is satisfied. Fig. 5.2shows the eigenvalues of B Up for different values of λ ∈ [0, 1].Convergence estimatesFinally, we summarize the convergence issues in a theorem. Taking into accountthe consistency estimate (5.10), as well as the monotonicity condition (5.13), whichalso ensures l ∞ stability, we have at last the followingTheorem 5.2. Let f, g ∈ W 1,∞ , u be the solution of (5.2) and vjn be defined by(5.9). Then, for any j ∈ Z and n ∈ [1, T/∆t],∣∣v j n − u(x j , t n ) ∣ → 0 (5.14)as ∆t, ∆x → 0, with the CFL condition (5.13).Moreover, if u has bounded second derivatives in space and time, then‖V n − U(t n )‖ ∞≤ C∆x. (5.15)Numerical viscosityIn order to derive the modified equation, we start from the Taylor expansionsv(x j , t n+1 ) = v + ∆tv t + ∆t22 v tt + O(∆t 3 )v(x j−1 , t n ) = v − ∆xv x + ∆x22 v xx + O(∆x 3 ),

✐✐108 Chapter 5. First-order approximation schemes10.80.60.40.20−0.2−0.4−0.6−0.8−1−1 −0.5 0 0.5 1Figure 5.2. Eigenvalues of the upwind scheme for λ = 0.2, 0.6, 1 and 50 nodes.where we have kept the convention of omitting the arguments for functions computedat (x j , t n ). Plugging the expansions in the form (5.6) of the scheme, weobtainv+∆tv t + ∆t22 v tt+O(∆t 3 ) = c∆t] ([v − ∆xv x + ∆x2∆x2 v xx + O(∆x 3 ) + 1 − c∆t )v∆xand hence, simplifying the terms in v, dividing by ∆t and taking into account thatdiscretization steps vanish at the same rate,v t + cv x = 1 2 (c∆xv xx − ∆tv tt ) + O(∆x 2 ). (5.16)On the other hand, deriving this equation first with respect to t, then with respectto x, and eliminating the mixed term v xt , we obtainwhich, plugged into (5.16), gives at lastv tt = c 2 v xx + O(∆x), (5.17)v t + cv x = c∆x2 (1 − λ)v xx + O(∆x 2 ). (5.18)5.1.2 Central discretization: the Lax–Friedrichs schemeThe consistency rate of the upwind scheme is mostly limited by the fact that, usingonly two points in discretizing the space derivative, the approximation cannot gobeyond the first order. An attempt to increase the consistency rate would thereforeimply the use of a larger stencil of points. In the class of central schemes, thisis typically accomplished by enlarging the stencil symmetrically. As we will see,usually this enlargement also causes a more dispersive behaviour of the method.

✐✐5.1. Treating the advection equation 109Construction of the schemeThe simplest situation for a symmetric stencil is the use of x j−1 , x j e x j+1 . In thiscase, the space derivative u x (x j , t) can be approximated asu x (x j , t) = u(x j+1, t) − u(x j−1 , t)2∆x+ O(∆x 2 ). (5.19)In principle, this larger stencil allows to increase accuracy, although this result isnot achieved in the Lax–Friedrichs scheme.The constant coefficient equation A first, naive way of using (5.19) into (5.1)would lead to the semi-discrete scheme˙v j (t) + c v j+1(t) − v j−1 (t)2∆x= 0.However, an explicit one-step time discretization of this scheme, and in particularthe one obtained by applying the forward Euler scheme,v n+1j= vj n − c∆t [vn2∆x j+1 − vj−1n ],results in an unstable scheme. The classical way to stabilize the scheme is either toturn to a midpoint two-step time approximation (this leads to the so-called leapfrogscheme), or to modify it in the Lax–Friedrichs formv n+1j = vn j−1 + vn j+1− c∆t [vn2 2∆x j+1 − vj−1n ], (5.20)which clearly cannot be derived via the method of lines, although it also falls inthe class of conservative schemes. In terms of the parameter λ = c∆t/∆x, Lax–Friedrichs scheme may also be rewritten asv n+1j= 1 + λ2vj−1 n + 1 − λ v n2j+1. (5.21)Note that, in this case, no assumption on the sign of c is necessary.The variable coefficient equation Passing to the variable coefficient equation iseasier in Lax–Friedrichs scheme, since it is no longer required to take into accountthe sign of the advection term. By analogy with the upwind scheme, with a straightforwardmodification of the constant coefficient case we finally obtain the formv n+1j = vn j−1 + vn j+1− f(x j , t n ) ∆t [vn22∆x j+1 − vj−1n ]+ ∆tg(xj , t n ). (5.22)ConsistencyBefore applying the definition of consistency, let us remark that, for a smoothsolution u(x, t), by elementary interpolation arguments we haveu(x j−1 , t n ) + u(x j+1 , t n )2= u(x j , t n ) + O(∆x 2 ). (5.23)

✐✐110 Chapter 5. First-order approximation schemesThen, applying the Lax–Friedrichs scheme (5.22) to a solution and using (5.23), weestimate a component of the consistency error (4.9) asL LFj (∆; t, U(t)) = 1 [u(x j , t + ∆t) − u(x j , t) + O(∆x 2 ) +∆t+f(x j , t) ∆t]2∆x (u(x j+1, t) − u(x j−1 , t)) − ∆tg(x j , t) == u(x ( )j, t + ∆t) − u(x j , t) ∆x2+ O +∆t∆t+f(x j , t) u(x j+1, t) − u(x j−1 , t)− g(x j , t) =2∆x( ) ∆x2= u t (x j , t) + O(∆t) + O +∆t+f(x j , t)u x (x j , t) + O(∆x 2 ) − g(x j , t) =( ) ∆x2= O(∆t) + O∆twhere only the relevant terms have been considered. The consistency estimate inHölder norm for the Lax–Friedrichs scheme reads therefore∥ L LF (∆; t, U(t)) ∥ ( )≤ C ∆t + ∆x2 . (5.24)∆tRemark 5.3. Note that, as in the case of the upwind scheme, the balance of thetwo terms requires steps of the same order of magnitude, and this choice is also finefor stability reasons. However, in the case of the LF scheme, setting ∆t of the orderof ∆x 2 or smaller would result in a nonconsistent scheme.StabilityAnalysis of the Lax–Friedrichs scheme also leads to stability conditions which coincideamong the various stability frameworks.CFL condition Referring again to the constant coefficient case, and to the schemein the version (5.21), the numerical solution v n+1j at (x j , t n+1 ) depends on vj−1 n andvj+1 n . This causes a numerical domain of dependence in the form shown in Fig. 5.3.Note that the triangle is symmetrical (so that no special care should be taken aboutthe sign of the velocity c) and enlarges itself at the same rate as for the upwindscheme, so that we satisfy the CFL condition under the same relationship|λ| = |c|∆t∆x≤ 1. (5.25)As in the case of the upwind scheme, the condition may be extended to variablecoefficient equations by replacing the constant advection speed c with the norm‖f‖ ∞ , so that the requirement becomes‖f‖ ∞ ∆t∆x≤ 1. (5.26)

✐✐5.1. Treating the advection equation 111Figure 5.3. Numerical domain of dependence for the Lax–Friedrichs scheme.Monotonicity We apply again the general condition for the monotonicity of alinear scheme. In the the Lax–Friedrichs scheme, the only nonzero entries on thej–th row areb LFj,j−1(∆; t n ) = 1 (1 + f(x j , t n ) ∆t )2∆xb LFj,j+1(∆; t n ) = 1 (1 − f(x j , t n ) ∆t ).2∆xOnce again, they have unity sum, and one of the two is necessarily positive. Therefore,the scheme is invariant for the addition of constants, and is monotone providedthe other term is positive for any j. This leads to the same condition obtained forthe upwind scheme, that is‖f‖ ∞ ∆t≤ 1. (5.27)∆xVon Neumann analysis We refer here to the scheme in the form (5.21). Given aneigenvector V of the scheme, an eigenvalue ρ satisfies the conditionρv j = 1 + λ2that is, using the form v j = e ijω ,v j−1 + 1 − λ v j+12ρe ijω = 1 + λ e i(j−1)ω + 1 − λ e i(j+1)ω22which gives the eigenvalue ρ asρ = 1 + λ e −iω + 1 − λ e iω =22= cos ω − iλ sin ω.In practice, eigenvalues are located on an ellipse centered at the origin, with unityhorizontal semi-axis and vertical semi-axis λ. The ellipse itself and the eigenvaluesare therefore contained in the unity disc if λ ∈ [0, 1]. Fig. 5.4 shows the eigenvaluesof B LF for different values of λ.

✐✐112 Chapter 5. First-order approximation schemes10.80.60.40.20−0.2−0.4−0.6−0.8−1−1 −0.5 0 0.5 1Figure 5.4. Eigenvalues of the Lax–Friedrichs scheme for λ = 0.2, 0.6, 1and 50 nodes.Convergence estimatesAgain, we summarize the convergence study in a theorem. Consistency followsfrom (5.24), once assumed that ∆x = o(∆t 1/2 ), and stability from the monotonicitycondition (5.27). As for the upwid scheme, the smoothness required on the solutionto achieve first-order convergence is to have bounded second derivatives in spaceand time, but also that ∆x ∼ ∆t. We can therefore state the convergence result as:Theorem 5.4. Let f, g ∈ W 1,∞ , u be the solution of (5.2) and vjn be defined by(5.22). Then, for any j ∈ Z and n ∈ [1, T/∆t],∣ vnj − u(x j , t n ) ∣ → 0 (5.28)as ∆t, ∆x → 0, with ∆x = o(∆t 1/2 ) and the CFL condition (5.27).Moreover, if u has bounded second derivatives in space and time and ∆x = c∆t forsome constant c, then‖V n − U(t n )‖ ∞≤ C∆x. (5.29)Numerical viscosityFirst, we compute the Taylor expansionsv(x j , t n+1 ) = v + ∆tv t + ∆t22 v tt + O(∆t 3 )v(x j−1 , t n ) = v − ∆xv x + ∆x22 v xx + O(∆x 3 )v(x j+1 , t n ) = v + ∆xv x + ∆x22 v xx + O(∆x 3 )

✐✐5.1. Treating the advection equation 113(again, we have omitted the arguments for functions computed at (x j , t n )). Usingsuch expansions in (5.20), we havev + ∆tv t + ∆t22 v tt + O(∆t 3 ) = v + ∆x22 v xx + O(∆x 3 ) − c∆t [2∆xvx + O(∆x 3 ) ]2∆xso that by simplifying, dividing by ∆t and using again the fact that discretizationsteps vanish at the same rate, we getv t + cv x = ∆x22∆t v xx − ∆t2 v tt + O(∆x 2 ), (5.30)and using again (5.17) (which can be derived in the same way for (5.30)), we obtainat lastv t + cv x = 1 ( )∆x22 ∆t − c2 ∆t v xx + O(∆x 2 ). (5.31)Comparing the expression of the numerical viscosity with the corresponding expressionfor the upwind scheme, it is apparent that the Lax–Friedrichs scheme has amore viscous behaviour. In fact, taking into account that |λ| ≤ 1,( )1 ∆x22 ∆t − c2 ∆t = |c|∆x ( ) 12 |λ| − |λ|≥ |c|∆x (1 − |λ|)2so that the numerical viscosity is greater in the LF than in the upwind scheme. Forexample, if |λ| = 1/2, then the viscosity of the upwind scheme is |c|∆x/4, whereasfor the LF scheme we get a viscosity of 3|c|∆x/4 (this is not surprising, whenconsidering that the numerical domain of dependence is twice as large). Note alsothat in the LF scheme the numerical viscosity is unbounded for vanishing Courantnumbers. This is related to the loss of consistency pointed out in remark 5.3.5.1.3 Semi–Lagrangian discretization: theCourant–Isaacson–Rees schemeIn many respects, Semi-Lagrangian schemes are upwind schemes. They have beenfirst proposed in the field of hyperbolic systems by Courant, Isaacson and Rees in[CIR52] in a form which precisely gives the first order upwind scheme when appliedto the advection equation. The possibility of making them work at large Courantnumbers has been recognized later, as soon as they have been applied to problemswith a relevant computational complexity.Construction of the schemeThe construction of the Courant–Isaacson–Rees scheme is performed by a discretizationon the representation formula (1.6), (1.7), rather then on the equation itself.Following this strategy, an intermediate semi-discrete stage would be continuous inspace, discrete in time. As in the previous examples, we start from the constantcoefficient case, then turn to the more general case.The constant coefficient equation In this case, the representation formula forthe solution takes the simpler form (1.3). Note that we are interested in making

✐✐114 Chapter 5. First-order approximation schemesthe scheme evolve on a single time step, from the time t n to t n+1 , and to assign avalue to a given node x j . Therefore, (1.3) should be rewritten asu(x j , t n+1 ) = u(x j − c∆t, t n ). (5.32)We point out that, unless for computing the solution only at discrete points of thespace-time grid, no numerical approximation has been introduced yet. However,since the point x j − c∆t is not in general a grid point, the unknown value u(x j −c∆t, t n ) has to be replaced by a numerical reconstruction obtained by interpolationbetween node values. In the original, first-order version of the CIR scheme, this isperformed by a P 1 interpolation.We obtain therefore the numerical scheme:v n+1j = I 1 [V n ](x j − c∆t) (5.33)complemented with the initial condition v 0 j = u 0(x j ).In order to write the right-hand side more explicitly, assume first that c∆t < ∆x,so thatx j − c∆t ∈ (x j−1 , x j ].Setting now λ = c∆t/∆x, the P 1 interpolation of the sequence V n at this pointmay be written asI 1 [V n ](x j − c∆t) = λv n j−1 + (1 − λ)v n j ,so that the explicit form of (5.33) becomes:v n+1j = λv n j−1 + (1 − λ)v n j . (5.34)Note that, despite being obtained through a different procedure, the result preciselycoincides with the upwind discretization. But, although Courant, Isaacson and Reesoriginally proposed their scheme in this version, since the representation formula(5.32) holds for any ∆t, it is also possible to use Courant numbers beyond theunity, this meaning that the foot of the characteristic starting at x j is no longer inan adjacent cell. Denoting by ⌊·⌋ the integer part, any λ > 1 is split asλ = λ ′ + ⌊λ⌋and the cell in which the reconstruction is performed is ( x j−⌊λ⌋−1 , x j−⌊λ⌋]. Hence,for Courant numbers exceeding the unity, (5.33) takes the formv n+1j = λ ′ v n j−⌊λ⌋−1 + (1 − λ′ )v n j−⌊λ⌋ . (5.35)The variable coefficient equation In the more general situation of (5.2), we startfrom the representation formula in the form (1.6), (1.7), written at the node x j andon a single time step:u(x j , t n+1 ) =∫ tn+1t ng(y(x j , t n+1 ; s), s)ds + u(y(x j , t n+1 ; t n ), t n ) (5.36)where y(x, t; s) solves (1.7).In (5.36), in addition to the reconstruction of the value of u(y(x j , t n+1 ; t n ), t n ), twomore approximations are required. First, the point y(x j , t n+1 ; t n ) itself should be

✐✐5.1. Treating the advection equation 115approximated, since the exact solution of (1.7) is not known. Second, the integralshould be evaluated by some quadrature formula. In the simplest situation, thepoint y(x j , t n+1 ; t n ) is replaced by its Euler approximation,x j − ∆tf(x j , t n+1 ) ≈ y(x j , t n+1 ; t n ), (5.37)and the integral in (5.36) by the rectangle quadrature on [t n , t n+1 ],∆tg(x j , t n+1 ) ≈∫ tn+1t ng(y(x j , t n+1 ; s), s)ds. (5.38)Plugging now (5.37), (5.38) and the P 1 reconstruction of u into (5.36), and incorporatingthe initial condition, we obtain a scheme in the form{v n+1j = ∆tg(x j , t n+1 ) + I 1 [V n ](x j − ∆tf(x j , t n+1 ))vj 0 = u (5.39)0(x j ).ConsistencyIn analysing consistency of the CIR scheme, instead of using the equation, we willrather compare the scheme with the representation formula (1.6), (1.7). LetS CIRj (∆; t, V ) = ∆tg(x j , t + ∆t) − I 1 [V ](x j − ∆tf(x j , t + ∆t)). (5.40)First, note that under smooth data, the elementary approximations introduced inthe scheme satisfy the error estimates:|x j − ∆tf(x j , t + ∆t) − y(x j , t + ∆t; t)| = O(∆t 2 ), (5.41)∫ t+∆t∣ ∆tg(x j, t + ∆t) − g(y(x j , t + ∆t; s), s)ds∣ = O(∆t2 ), (5.42)t‖u − I 1 [U]‖ ∞= O(∆x 2 ). (5.43)Bounds (5.41)–(5.43) are classical for respectively the Euler scheme, the rectanglequadrature and the P 1 interpolation. Writing now u(x j , t + ∆t) by means of (5.36),we may now estimate the left-hand side of (5.44) as∣∣L CIRj(∆; t, U(t)) ∣ = 1 ∣∣u(x j , t + ∆t) − SjCIR (∆; t, U(t)) ∣ ≤∆t≤ 1 ∫ t+∆tg(y(x∆t∣j , t + ∆t; s), s)ds + u(y(x j , t + ∆t; t), t) −t−∆tg(x j , t + ∆t) − I 1 [U(t)](x j − ∆tf(x j , t + ∆t))∣ ≤[∣≤ 1 ∣∣∣∣ ∫ t+∆tg(y(x j , t + ∆t; s), s)ds − ∆tg(x j , t + ∆t)∆t∣ +t+ |u(y(x j , t + ∆t; t), t) − I 1 [U(t)](y(x j , t + ∆t; t))| ++ |I 1 [U(t)](y(x j , t + ∆t; t)) − I 1 [U(t)](x j − ∆tf(x j , t + ∆t))|].

✐✐116 Chapter 5. First-order approximation schemesThe first two terms in the last display can be directly estimated by (5.42) and(5.43), whereas for the third we note that the Lipschitz constant of the reconstructedsolution I 1 [U] is no larger than the Lipschitz constant L u of u, so that applyingagain (5.41) we get∣ LCIRj (∆; t, U(t)) ∣ 1 [≤ O(∆t 2 ) + O(∆x 2 ) + L u O(∆t 2 ) ]∆twhich in turn implies the estimate∥ L CIR (∆; t, U(t)) ∥ ( )≤ C ∆t + ∆x2 . (5.44)∆tThis consistency estimate is similar to the corresponding estimate for the Lax–Friedrichs scheme (consistency requires that ∆x 2 = o(∆t) and its order is maximizedwhen ∆t ∼ ∆x). However, when taking into account that the CIR scheme turnsout to be unconditionally stable, and hence not restricted to operate with stepsof the same order of magnitude, the interplay between the two discretization stepsbecomes less trivial. This effect will be analysed in detail later in this section, andagain in chapter 6.StabilityThe main issue in the stability analysis of the CIR scheme (and of SL schemes ingeneral) is its unconditional stability, which allows for large Courant numbers. Weanalyse this feature of the scheme from different viewpoints.CFL condition The fulfillment of the CFL condition is obtained in the CIR schemeby shifting the reconstruction stencil along characteristics, in the neighbourhood ofthe point x j −c∆t. This causes some form of self-adaptation of the numerical domainof dependence, although at the additional cost of locating the foot of characteristics.Another advantage is the lower number of grid points involved, this generallyimplying a lower numerical viscosity. Figure 5.5 shows this point at a comparisonwith the corresponding situations for upwind and Lax–Friedrichs schemes (Figures5.1 and 5.3). In particular, an enlargement of the numerical domain of dependenceof O(∆x) at each step is multiplied by a number of steps of O(1/∆t), turning intoa numerical domain of dependence of radius O(∆x/∆t). If the refinement is madewith diverging Courant numbers (and this can really be the case in Semi-Lagrangianschemes), then this domain collapses towards the foot of the characteristic.Monotonicity Rewriting the interpolation operator I 1 [V ] in terms of the basisfunctions ψ [1]i , the CIR scheme takes the formSjCIR (∆; t n , V ) = ∆tg(x j , t n+1 ) − ∑ iv i ψ [1]i (x j − ∆tf(x j , t n+1 )) (5.45)so thatb CIRji (∆; t n ) = ψ [1]i (x j − ∆tf(x j , t n+1 )). (5.46)Now, since ψ [1]i (x) ≥ 0 and ∑ i ψ[1] i (x) ≡ 1 (in fact, the latter property is true forLagrange interpolation of any order), the CIR scheme is invariant for the additionof constants and monotone for any ∆x and ∆t.

✐✐5.1. Treating the advection equation 117Figure 5.5. Numerical domain of dependence for the CIR scheme.Von Neumann analysis As usual, we refer here to the scheme in the form (5.33).Given an eigenvector V of the scheme, in the form v j = e ijω , the condition whichcharacterizes ρ as an eigenvalue readsthat is,ρv j = λ ′ v j−⌊λ⌋−1 + (1 − λ ′ )v j−⌊λ⌋ ,ρe ijω = λ ′ e i(j−⌊λ⌋−1)ω + (1 − λ ′ )e i(j−⌊λ⌋)ωwhich allows to express the eigenvalue ρ asρ = e i⌊λ⌋ω [ λ ′ e −iω + (1 − λ ′ ) ] . (5.47)In the right-hand side of (5.47), the first is a pure phase term, (depending onlyon ⌊λ⌋) and is therefore irrelevant for stability, whereas the second, relevant termdepends only on λ ′ . Restricting therefore to the second term, and rearranging(5.47), we obtain (unless for replacing λ with λ ′ ) the same conclusions as for theupwind scheme, that is|ρ| = |(1 − λ ′ ) + λ ′ cos ω + iλ ′ sin ω| (5.48)Therefore, if λ ∈ [0, 1], eigenvalues are located on a circle centered at 1 − λ andwith radius λ (here, the CIR scheme coincides with the upwind scheme). If λ > 1,a further phase shift is superimposed on this curve. In both cases, λ ′ ∈ [0, 1] andthe Von Neumann condition is satisfied. Figure 5.6 shows the eigenvalues of B CIRfor two different values of λ having the same fractional part.Convergence estimatesFinally, we summarize this convergence study. We point out that (5.41), (5.42)require the functions f and g to be in W 1,∞ , whereas (5.43) requires that thesolution u is of W 2,∞ . Moreover, if u ∈ W 1,∞ , then (5.43) might be replaced by‖u − I[U]‖ ∞≤ C u ∆x.

✐✐118 Chapter 5. First-order approximation schemes10.80.60.40.20−0.2−0.4−0.6−0.8−1−1 −0.5 0 0.5 1Figure 5.6. Eigenvalues of the CIR scheme for λ = 0.25, 2.25 and 50 nodes.Taking into account the conditional consistency of the scheme, we have at last thefollowingTheorem 5.5. Let f, g ∈ W 1,∞ , u be the solution of (5.2) and vjn be defined by(5.39). Then, for any j ∈ Z and n ∈ [1, T/∆t],∣ vnj − u(x j , t n ) ∣ → 0 (5.49)as ∆t → 0, ∆x = o ( ∆t 1/2) .Moreover, if u ∈ L ∞ ([0, T ], W s,∞ (R)) (s = 1, 2), then‖V n − U(t n )‖ ∞≤ C)(∆t + ∆xs . (5.50)∆tNumerical viscosityTo compute the numerical viscosity of the CIR scheme, we first plug a regularextension of the numerical solution into the scheme and use the estimate for P 1Lagrange interpolation error to express the value I 1 [V n ](x j − c∆t) asI 1 [V n ](x j − c∆t) = v(x j − c∆t, t n ) − λ′ (λ ′ − 1)∆x 2v xx (ξ j ),2with ξ j an unknown point located in the same interval of x j −c∆t. Next, taking intoaccount that the computation of the second derivative in the interpolation error isperformed at a point which differs from x j − c∆t by an O(∆x), we get from(5.33):v(x j , t n+1 ) = v(x j − c∆t, t n ) + λ′ (1 − λ ′ )∆x 2[v xx (x j − c∆t) + O(∆x)]2

✐✐5.1. Treating the advection equation 119which is in the form (4.85), with k = 2, once set the numerical viscosity asν = λ′ (1 − λ ′ )∆x 2.2∆tThe viscosity coefficient is positive, depends on λ ′ , and vanishes if λ ′ = 0 (this isconsistent with the fact that if the feet of characteristics coincide with grid nodes,the solution is advected without errors). In order to carry out a worst-case analysis,we note that its largest value is attained for λ ′ = 1/2. We can then deduce that,in the worst case, the numerical solution corresponding to a local interpolation offirst order solves the modified equationv t + cv x = ∆x28∆t v xx + o( ∆x2∆t). (5.51)Note that the viscous term originates from the accumulation of interpolation errors,and can be reduced by using large Courant numbers.High–order characteristics tracking in the CIR schemeThe first-order discretization in (5.32) is suitable for the homogeneous, constant coefficientcase, where it provides the exact upwinding along characteristics. However,in nonhomogeneous problems, as well as in the variable coefficient case, it could leadto an undesired error in time discretization. In practice, in order to balance the twoerror terms, the scheme could be forced again to work at small Courant numbers(hence, with an unnecessary computational complexity), in spite of its unconditionalstability for large time steps. Another good reason to work at large time steps isthe reduction of numerical viscosity, as it has just been noticed. Therefore, a moreaccurate approximation of characteristics and integral is desirable, and in fact canbe incorporated in the scheme without changing its stability properties.Looking at the proof of consistency, it is clear that the consistency rates in (5.41)and (5.42) should be increased in parallel, the lower of the two being the relevantrate in the first term of the global estimate (5.44). Following [FF98], a general wayto do this is to introduce the augmented system:(ẏ(x, ) ( )t; s) f(y(x, t; s), s)=(5.52)˙γ(x, t; s) −g(y(x, t; s), s)with the initial conditions y(x, t; t) = x, γ(x, t; t) = 0. In what follows, we denoteby respectively X ∆ (x, t; s) and G ∆ (x, t; s) the approximations of y(x, t; s) andγ(x, t; s), where typically x = x j , t = t n+1 and s = t n . For example, in the classicalCIR scheme (5.39) the Euler/rectangle rule discretization used corresponds to theapplication of the Euler scheme to (5.52), so that( )X ∆ (x j , t n+1 ; t n )G ∆ =(x j , t n+1 ; t n )( ) ( )xj f(xj , t− ∆tn+1 )0 −g(x j , t n+1 )With this modifications, the scheme (5.39) can be rewritten more generally as{v n+1j = G ∆ (x j , t n+1 ; t n ) + I 1 [V n ] ( X ∆ (x j , t n+1 ; t n ) )v 0 j = u 0(x j ).(5.53)(5.54)

✐✐120 Chapter 5. First-order approximation schemesAssuming now that (5.41), (5.42) are replaced by the respective estimates of orderp, ∣ ∣ X ∆ (x j , t n+1 ; t n ) − y(x j , t n+1 ; t n ) ∣ ∣ = O(∆t p+1 ), (5.55)∣ G∆ (x j , t n+1 ; t n ) −∫ tn+1t ng(y(x j , t n+1 ; s), s)ds∣ = O(∆tp+1 ), (5.56)and retracing the proof of (5.44), we get a consistency error bounded by∥ L CIR (∆; t, U(t)) ∥ ( )≤ C ∆t p + ∆x2 . (5.57)∆tUsually, for (5.55), (5.56) to hold it is necessary to assume at least that f, g ∈ W p,∞ .Since it is also immediate to check that monotonicity still holds for the scheme inthe version (5.54), we can reformulate Theorem 5.5 in the more general formTheorem 5.6. Let f, g ∈ W p,∞ , u be the solution of (5.2) and v n j be definedby (5.54). Assume moreover that (5.55), (5.56) hold. Then, for any j ∈ Z andn ∈ [1, T/∆t], ∣ ∣ v n j − u(x j , t n ) ∣ ∣ → 0 (5.58)as ∆t → 0, ∆x = o ( ∆t 1/2) .Moreover, if u ∈ L ∞ ([0, T ], W s,∞ (R)) (s = 1, 2), then)‖V n − U(t n )‖ ∞≤ C(∆t p + ∆xs . (5.59)∆tNote that, in comparison with the fully first-order CIR scheme, in this latterversion the terms of the consistency error are balanced under increasing Courantnumbers if p > 1. Introducing the relationship ∆t = ∆x α , the consistency error isO ( ∆x αp + ∆x 2−α) and its order is maximal with the choice α = 2/(p + 1) whichbalances the two terms. For example, using a second-order scheme in (5.60), wewould have an optimal relationship for α = 2/3.Examples: one-step schemes Applying a one-step scheme to (5.52), and restrictingto a single backward time step, we have:( ) ( ( (X ∆ (x, t; t − ∆t) x Φf −∆t; x, t, XG ∆ = − ∆t(x, t; t − ∆t) 0)∆ (x, t; t − ∆t) ) )(Φ −g −∆t; x, t, X ∆ (x, t; t − ∆t) ) (5.60)where the function Φ is split according to (5.52), and X ∆ is understood as thesolution of a system in implicit schemes, in which the right-hand side of (5.60)depends genuinely on X ∆ itself. Table 5.1 puts in the form (5.60) the same examplesof table 3.1.In these examples, we have taken for granted that the functions f and g can becomputed for any generic argument, that is, that they have an explicit expression.In a number of situation, like the nonlinear advection in fluid dynamics models, itcould rather happen that• both the advection and the source term result from previous computationsand/or physical measures, and are only known at grid nodes;

✐✐5.1. Treating the advection equation 121scheme form orderFE(Φ ) f −∆t; x, t, X∆= f(x, t) p = 1(Φ ) −g −∆t; x, t, X∆= −g(x, t)BE Φ f(−∆t; x, t, X∆ ) = f ( X ∆ , t − ∆t ) p = 1Φ −g(−∆t; x, t, X∆ ) = −g ( X ∆ , t − ∆t )(H Φ ) f −∆t; x, t, X∆= 1 2[f(x, t) + f(x − ∆tf(x, t), t − ∆t)] p = 2(Φ ) −g −∆t; x, t, X∆= − 1 2[g(x, t) + g(x − ∆tf(x, t), t − ∆t)]CN(Φ ) [ (f −∆t; x, t, X∆= 1 2 f(x, t) + f X ∆ , t − ∆t )] p = 2(Φ ) [ (−g −∆t; x, t, X∆= − 1 2 g(x, t) + g X ∆ , t − ∆t )]Table 5.1. First- and second-order one-step schemes for the augmentedsystem (5.52)• the advection term is known at previous time steps, but not yet at the stept n+1 .The first situation requires that f and g be interpolated to be computedat a generic point. Assuming that f is interpolated with degree q and thereforereconstructed with error O(∆x q+1 ), the accuracy assumption (5.55) changes into∣ X ∆ (x, t; t − ∆t) − y(x, t; t − ∆t) ∣ ∣ ≤ O(∆tp+1 ) + O ( ∆t∆x q+1) ,and the consistency error into∥ L CIR (∆; t, U(t)) ∥ ()≤ C ∆t p + ∆x q+1 + ∆x2∆t(clearly, the same ideas also apply to the source term g). For example, if a secondorderscheme is used to move along characteristics, the optimal relationship withoutinterpolation for f would be, as we said above, ∆x = ∆t 3/2 . Introducing theinterpolation of f, we have in turn∥∥L CIR (∆; t, U(t)) ∥ (≤ C ∆t 2 + ∆t 3 (q+1)) 2 .It suffices therefore to interpolate f with degree q = 1 to preserve the consistencyrate obtained without interpolation. Note also that in case a constant Courantnumber is kept (∆x ∼ ∆t), the consistency rate is preserved if q + 1 ≥ p, so a linearinterpolation is still enough for a second-order time discretization.

✐✐122 Chapter 5. First-order approximation schemesThe second situation, that is, f being unknown at the (n + 1)–th time step, isusually unsuitable to be treated with the one-step schemes of table 5.1, unless witha first order approximation. Since under minimal smoothness assumptions we havef(x, t) = f(x, t − ∆t) + O(∆t),g(x, t) = g(x, t − ∆t) + O(∆t),then, replacing the values of f and g computed at t in the scheme (5.39) with thesame values computed at t − ∆t, we would obtain for X ∆X ∆ (x, t; t − ∆t) = x − ∆tf(x, t − ∆t) + O ( ∆t 2)(a similar error holds for G ∆ ) so that X ∆ , G ∆ are computed using the values atthe previous time step and without any loss in the consistency rate. An alternativechoice is given by the backward Euler scheme, which only requires the knowledge off at t − ∆t. In this case, the foot of characteristic is located by solving the systemX ∆ (x, t; t − ∆t) = x − ∆tf ( X ∆ (x, t; t − ∆t), t − ∆t ) ,whereas the source term is given by G ∆ (x, t; t − ∆t) = g(X ∆ (x, t; t − ∆t), t − ∆t).The system is already in fixed-point form, and could be solved for X ∆ by theiterationXk+1 ∆ = x − ∆tf ( Xk ∆ , t − ∆t )provided the right-hand side is a contraction, that is, provided∆t < 1‖J f ‖ ,J f denoting the Jacobian matrix of the vectorfield f. Note that this introduces abound on ∆t, but this bound depends on the derivatives of the advection speedrather than on its magnitude. Note also that the method is first-order, so thatit gives no improvement in accuracy over the explicit Euler method used in theclassical CIR scheme.Examples: multistep schemes The most classical application of multistep schemesfor the approximation of the foot of characteristics involves the midpoint method(see table 3.1). In this case, the equation of characteristics is integrated 2∆t backin time, and more precisely X ∆ (x j , 2∆t) solves the fixed-point system( x + XX ∆ ∆ )(x, t; t − 2∆t)(x, t; t − 2∆t) = x − 2∆tf, t − ∆t . (5.61)2Note that, according to the definition of the midpoint method, the vector fieldf should be computed at the point X ∆ (x, t; t − ∆t). Along a smooth trajectory,however, we haveX ∆ (x, t; t − ∆t) = x + X∆ (x, t; t − 2∆t)2+ O(∆t 2 ),so that by the same arguments used for the explicit Euler method, the approximatemidpoint method (5.61) still results in a second-order scheme. Accordingly, thesource term is computed as( x + XG ∆ ∆ )(x, t; t − 2∆t)(x, t; t − 2∆t) = 2∆tg, t − ∆t .2

✐✐5.1. Treating the advection equation 123Thus, the scheme would compute the approximate solution at time t n+1 by advectingthe approximate solution at t n−1 along the vectorfield computed at t n (such astructure is often referred to as a three-time-level scheme). In general, the latterfixed-point system can also be solved iteratively under the same time-step limitationas for the backward Euler scheme, and we expect that in general the value of f atthe right-hand side should be obtained by interpolation. Since the resulting schemeis second-order in time, as remarked above, linear interpolation on f is enough topreserve the consistency rate.5.1.4 Multiple space dimensionsWe briefly sketch here how the techniques introduced in the previous sections canbe adapted to treat more general cases, in particular problems in higher dimensionand posed on bounded domains.To treat the case of multiple dimensions, it is convenient to rewrite the advectionequation,u t + f(x, t) · Du = g(x, t) R d × [0, T ],in the equivalent formu t +d∑f i (x, t)u xi = g(x, t) R d × [0, T ]. (5.62)i=1For simplicity, we review the various approaches referring to the two-dimensionalcase. Once given the general form, we briefly discuss consistency and monotonicity.Upwind scheme The upwind scheme may be better explained referring to thehomogeneous, constant coefficient equationu t + c 1 u x1 + c 2 u x2 = 0.Given that the consistency of the scheme is insensitive to the choice of taking rightor left incremental ratios, the crucial point is to obtain a stable (in particular,monotone) scheme. Keeping the strategy of approximating space derivatives by leftincremental ratios, we would obtainv n+1j 1,j 2− vj n 1,j 2vj n + c 1,j 2− vj n 1−1,j 2vj n1 + c 1,j 2− vj n 1,j 2−12 = 0,∆t∆x 1∆x 2which is clearly consistent. Solving this relationship for v n+1j 1,j 2, we obtainv n+1j 1,j 2= vj n 1,j 2− c 1∆t [vn∆x j1,j 2− v n ] c 2 ∆t [j 1−1,j 2 − vn1 ∆x j1,j 2− v n ]j 1,j 2−1 =(2= 1 − c 1∆t− c )2∆tvj n ∆x 1 ∆x 1,j 2+ c 1∆tvj n2 ∆x 1−1,j 2+ c 2∆tvj n1 ∆x 1,j22−1,and it is immediate to see that all coefficients are nonnegative (for partial Courantnumbers c i ∆t/∆x i small enough) if and only of c 1 , c 2 ≥ 0. This suggests thateach of the terms f i (x, t)v xi should be discretized in an upwind form: with a leftincremental ratio if f i (x j , t n ) > 0, with a right incremental ratio if f i (x j , t n ) < 0.We give up the boring, but conceptually simple, description of the details.

✐✐124 Chapter 5. First-order approximation schemesLax–Friedrichs scheme In extending the Lax–Friedrichs scheme to (5.62), it shouldbe noted that the average which replaces vj n 1,j 2should include all points involvedin the finite difference approximations of space derivatives, and therefore in thiscase (d = 2) the four points vj n 1−1,j 2, vj n 1+1,j 2, vj n and 1,j 2−1 vn j 1,j 2+1 . This gives theschemev n+1 (j 1,j 2− 1 4 vnj1−1,j 2+ vj n 1+1,j 2+ vj n + 1,j 2−1 vn j 1,j 2+1)+∆t+f 1 (x j , t n ) vn j 1+1,j 2− v n j 1−1,j 2∆x 1+ f 2 (x j , t n ) vn j 1,j 2+1 − vn j 1,j 2−1∆x 2= g(x j , t n )(again, this discretization may be shown to be consistent using the same argumentsof the one-dimensional case), and solving for v n+1j 1,j 2we get:v n+1j 1,j 2= 1 (vn4 j1−1,j 2+ vj n 1+1,j 2+ vj n 1,j 2−1 + vj n )1,j 2+1 −−f 1 (x j , t n ) ∆t [vn∆x j1+1,j 2− v n ]j 1−1,j 2 −1−f 2 (x j , t n ) ∆t∆x 2[vnj1,j 2+1 − v n j 1,j 2−1]+ ∆tg(xj , t n ).In this way, the nonzero coefficients of the scheme are of the formb LFjk (∆; t n ) = 1 4 ± f i(x j , t n ) ∆t∆x i(for suitable values of the multiindex k and of the index i), and again can be madenonnegative by keeping the partial Courant numbers small enough.Courant–Isaacson–Rees scheme In more than one space dimension, the CIRscheme can still be written in a formally unchanged way, that is,v n+1j = G ∆ (x j , t n+1 ; t n ) + I 1 [V n ] ( X ∆ (x j , t n+1 ; t n ) ) .In this case, however, it should be understood that:• The augmented system (5.52) is now in dimension d + 1: the approximationof the source term remains scalar, whereas characteristics are curves in R d .Thus, X ∆ is now a point of R d itself.• The reconstruction I 1 [V ](·) is a multidimensional P 1 or Q 1 interpolation.The monotonicity of the scheme is proved following the same argument as ina single space dimension: we have in fact that the coefficients of the scheme aregiven byb CIRji (∆; t n ) = ψ [1] (i X ∆ (x j , t n+1 ; t n ) ) ,and are nonnegative if the basis functions for interpolation are nonnegative (inmultiple dimension, this is clearly true for tensorized linear bases and P 1 finiteelements).

✐✐5.1. Treating the advection equation 1255.1.5 Boundary conditionsThe basic theory has been outlined so far without any reference to boundary conditions,which naturally arise when posing a problem on a bounded domain. Asituation which allows to keep essentially the same framework of unbounded gridsis the case of periodic conditions given on a d-dimensional torus [0, P 1 ]×· · ·×[0, P d ].Here, the periodicity is imposed in classical difference schemes by identifying thecorresponding nodes (and values) on opposite sides of the torus, whereas in SLschemes a point outside the torus is identified with an internal point via the relationshipu(x 1 ± P 1 , . . . , x d ± P d ) = u(x 1 , . . . , x d ).The case of Dirichlet conditions is more interesting. Assume now that Ω ≢ R dand that Dirichlet data are imposed on the inflow part of the boundary ∂Ω:u(x, t) = b(x, t) (x, t) ∈ Γ in × [0, T ]. (5.63)We review the main points in the implementation of this boundary condition in theupwind, LF and CIR schemes, avoiding the most technical issues.Upwind scheme To fix ideas, assume that the computational domain is a rectangle.Due to the definition of the scheme, nodes on each side the boundary areassigned a value based on internal nodes if the vector field f points outwards, thatis, on the outflow boundary Γ out , and this agrees with the fact that Dirichlet conditionscannot be enforced on this portion of the boundary. Nodes on Γ in areassigned a value by (5.63), while nodes placed at the angles may be at the interfacebetween Γ in and Γ out . In this situation, not all the partial incremental ratios maybe computed using internal nodes, so on such a node we force the boundary condition(5.63). Clearly, more complex geometries require to treat a larger number ofpossible situations.Lax–Friedrichs scheme In the Lax–Friedrichs scheme, assigning a value to a nodealways requires that other nodes exist around it. So, while this strategy allows toassign a boundary value to a node in Γ in (since it needs not to be computed by thescheme), it does not allow to compute the values at nodes of Γ out , since it wouldrequire a further layer of nodes. On the other hand, if the boundary condition bis imposed on the whole of ∂Ω, what typically happens is the development of aboundary layer on Γ out . To get rid of this effect, a simple response would be totreat in upwind form the nodes on this part of the boundary.Courant–Isaacson–Rees scheme The CIR scheme requires a more complex treatmentof the Dirichlet condition. First, we rewrite the extended version (2.44) of therepresentation formula between t n and t n+1 :u(x j , t n+1 ) =∫ tn+1t n∨θ(x j,t n+1)g(y(x j , t n+1 ; s), s)ds ++u(y(x j , t n+1 ; t n ∨ θ(x j , t n+1 )), t n ∨ θ(x j , t n+1 )). (5.64)As we have discussed in Chapter 2, this amounts to single out a case in which thesolution is propagated from an internal point, and a case in which it propagates fromthe boundary. The numerical counterpart of this procedure requires to determine

✐✐126 Chapter 5. First-order approximation schemesthe point, if any exists, at which the discrete trajectory X ∆ crosses the boundaryof Ω. If we assume for simplicity that the boundary ∂Ω is defined as the zero levelset of a continuous function ϕ : R N → R, so that x ∈ Ω if and only if ϕ(x) < 0,then we can define a variable time step δ(x j , t n+1 ) as the minimum between ∆t andthe solution of the equationϕ ( X ∆ (x j , t n+1 ; t n+1 − δ(x j , t n+1 )) ) = 0.Note that we expect at most one solution in [t n , t n+1 ], for ∆t small enough and “nonpathological”sets Ω. Note also that, for consistency reasons, δ(x j , t n+1 ) should alsoreplace ∆t in the computation of X ∆ (e.g., in the construction of a one-step scheme).Once computed the variable step, we split the set of indices as I = I i ∪ I b ,so that δ(x j , t n+1 ) = ∆t for j ∈ I i (internal nodes), δ(x j , t n+1 ) ≤ ∆t for j ∈ I b(near-boundary nodes). Then, the CIR scheme is modified asv n+1j =⎧G ∆ (x j , t n+1 ; t n ) + I 1 [V n ] ( X ∆ (x j , t n+1 ; t n ) )⎪⎨if j ∈ I iG⎪⎩∆ (x j , t n+1 ; t n+1 − δ(x j , t n+1 ))++b ( X ∆ (x j , t n+1 ; t n+1 − δ(x j , t n+1 )), t n+1 − δ(x j , t n+1 ) ) if j ∈ I b .(5.65)5.1.6 ExamplesIn order to evaluate the schemes which have been presented in this section, we usethe simple one-dimensional example{u t (x, t) − (x − ¯x)u x (x, t) = 0 (x, t) ∈ (0, 1) × (0, 1)(5.66)u(x, 0) = u 0 (x)with three initial conditions u 0 of different regularity and with bounded support.In particular, the first is a bounded discontinuous function:whereas the second is Lipschitz continuous:and the third has bounded second derivative:u 0 (x) = 1 [0.25,0.5] (x), (5.67)u 0 (x) = max(1 − 16(x − 0.25) 2 , 0), (5.68)u 0 (x) = max(1 − 16(x − 0.25) 2 , 0) 2 . (5.69)In (5.66), characteristics are curved lines and this makes time discretization significantin SL schemes. Also, characteristics converge towards the point ¯x = 1.1, so thatthe initial condition is advected rightwards and the support shrinks for increasingtime.The test has been carried out with Upwind, Lax–Friedrichs and Courant–Isaacson–Rees schemes, the last being implemented both in its classical version(5.39) and in the version (5.54) with a second-order time discretization (denotedas CIR2). In the first three cases, grid refinement has been performed by keepinga constant ∆t/∆x relationship, and more precisely keeping the Courant number

✐✐5.1. Treating the advection equation 127L ∞ solutionn n Upwind LF CIR CIR225 1.93 · 10 −1 2.45 · 10 −1 1.06 · 10 −1 1.1 · 10 −150 1.66 · 10 −1 2.3 · 10 −1 1.16 · 10 −1 7.35 · 10 −2100 1.35 · 10 −1 2.2 · 10 −1 7.59 · 10 −2 6.31 · 10 −2200 1.1 · 10 −1 1.9 · 10 −1 6.64 · 10 −2 4.71 · 10 −2400 9.2 · 10 −2 1.56 · 10 −1 5.38 · 10 −2 4.01 · 10 −2rate 0.27 0.16 0.24 0.36W 1,∞ solutionn n Upwind LF CIR CIR225 1.55 · 10 −1 2.39 · 10 −1 1.29 · 10 −1 3.36 · 10 −250 1.09 · 10 −1 2.09 · 10 −1 7.62 · 10 −2 1.78 · 10 −2100 7.05 · 10 −2 1.72 · 10 −1 4.1 · 10 −2 8.43 · 10 −3200 4.39 · 10 −2 1.28 · 10 −1 2.21 · 10 −2 4.42 · 10 −3400 2.66 · 10 −2 8.68 · 10 −2 1.16 · 10 −2 1.99 · 10 −3rate 0.64 0.37 0.87 1.02W 2,∞ solutionn n Upwind LF CIR CIR225 1.52 · 10 −1 2.17 · 10 −1 1.32 · 10 −1 2.97 · 10 −250 1.14 · 10 −1 1.96 · 10 −1 7.52 · 10 −2 1.44 · 10 −2100 7.61 · 10 −2 1.68 · 10 −1 4.01 · 10 −2 6.09 · 10 −3200 4.63 · 10 −2 1.32 · 10 −1 2.08 · 10 −2 2.6 · 10 −3400 2.63 · 10 −2 9.27 · 10 −2 1.06 · 10 −2 1.05 · 10 −3rate 0.63 0.31 0.91 1.21Table 5.2. Errors in the 2-norm, Upwind, Lax–Friedrichs, Courant–Isaacson–Rees (with first- and second-order time discretization) schemes.less than 0.6 in Upwind and LF schemes, and less than 6 in CIR scheme. Inthe case of the second-order time discretization for the CIR scheme, we have usedthe relationship ∆t ∼ ∆x 2/3 (more precisely, ∆t ≈ 2∆x 2/3 ) which optimizes theconsistency rate.We list in Table 5.2 the numerical errors in the 2-norm for the various schemes,along with overall convergence rates computed on the extreme steps of the refinement.Figure 5.7 shows the behaviour of the schemes on the advection of thecharacteristic function (5.67) (the exact solution is shown in solid line).The apparent point in the comparison of the various schemes is numerical diffusion.The accuracy of the LF scheme is reduced by its highly diffusive behaviour,but ranging from the Upwind to the CIR2 scheme the convergence rate has no dramaticchange, although it is clear how the possibility of working at large Courantnumbers reduces the numerical diffusion. In terms of absolute accuracy, the firstorderCIR scheme is about twice as accurate as the Upwind scheme, and the increasein accuracy is even more apparent for the CIR2 scheme, especially as the solution

✐✐128 Chapter 5. First-order approximation schemesFigure 5.7. Numerical results for the advection of a characteristic functionobtained via Upwind (upper left), Lax–Friedrichs (upper right), Courant–Isaacson–Rees (lower left) and Courant–Isaacson–Rees with second-order time discretization(lower right) schemes, 200 nodes.gets smoother (here, the theoretical convergence rate would be 4/3 ≈ 1.33).5.2 Treating the convex HJ equationIn this section, we show how the schemes introduced can be adapted to the onedimensional,convex Hamilton–Jacobi equation{u t (x, t) + H(u x (x, t)) = 0 (x, t) ∈ R × [0, T ],(5.70)u(x, 0) = u 0 (x).We will make the standing assumption that H is convex, and that there existsα 0 ∈ R such that{H ′ (α) ≤ 0 if α ≤ α 0 ,H ′ (5.71)(α) ≥ 0 if α ≥ α 0 .We also define:M H ′(L) = max[−L,L] |H′ | (5.72)which corresponds to the maximum speed of propagation of a solution with Lipschitzconstant L.

✐✐5.2. Treating the convex HJ equation 1295.2.1 Upwind discretizationIn adapting the upwind scheme to the nonlinear case, it should be taken into considerationthat the speed of propagation of the solution is H ′ (v x ). While it is perfectlyclear how to construct an upwind scheme for a speed of constant sign, care shouldbe taken at points where the speed changes sign, in order to obtain a monotonescheme.Construction of the schemeThe construction outlined will follow the guidelines of [CL84], in which monotoneschemes for HJ equations are derived from monotone schemes for conservation laws,and theory is carried out accordingly. The differenced form of the upwind schemeis (4.53),v n+1j = v n j − ∆tH Up (D j−1 [V n ], D j [V n ]) , (5.73)where the numerical Hamiltonian H Up is defined by⎧H(α) if α, β ≥ α 0 ,⎪⎨H Up H(β) + H(α) − H(α 0 ) if α ≥ α 0 , β ≤ α 0 ,(α, β) =H(α 0 ) if α ≤ α 0 , β ≥ α 0 ,⎪⎩H(β) if α, β ≤ α 0 .(5.74)Note that the situation in which speed changes sign is subject to a different handling,depending on the fact that characteristics converge or diverge.ConsistencySince the scheme is in differenced form, it actually suffices to apply proposition 4.18.If α = β = a, then the numerical hamiltonian (5.74) satisfiesH Up (a, a) = H(a), (5.75)and the consistency condition (4.51) is satisfied. Note that, in (5.74), the secondand third case only occur if a = α 0 .StabilityBy construction, schemes in differenced form (in particular, upwind and Lax–Friedrichs in what follows) are necessarily invariant for the addition of constants.Therefore, the main stability issues in this context will be CFL condition and monotonicity.CFL condition Since the maximum speed of propagation of the solution is M H ′(L),the condition which keeps characteristics within the numerical domain of dependenceisM H ′(L)∆t≤ 1. (5.76)∆xIn this case, this restriction is really only necessary – as we will see, monotonicityrequires a stronger condition.

✐✐130 Chapter 5. First-order approximation schemesMonotonicityscheme as∂S Upj∂v iFirst, we write the partial derivative of the j–th component of the[ ]∂HUp∂D j−1 [V ](∆; V ) = δ ij − ∆t+ ∂HUp ∂D j [V ]∂α ∂v i ∂β ∂v i(5.77)where α and β are the dummy variables used in the definition (5.74), and δ ij is theKronecker symbol. It is clear that⎧1∂D j−1 [V ]⎪⎨ ∆xif i = j= − 1∂v∆xif i = j − 1i ⎪⎩0 otherwise⎧1∂D j [V ]⎪⎨ ∆xif i = j + 1= − 1∂v∆xif i = ji ⎪⎩0 otherwise,so that, substituting into (5.77), we obtain the more explicit form∂∂v iS Upj (∆; V ) =⎧[ ⎪⎨1 − ∆t−∆t ∂HUp ∂D j−1[V ]∂α∂H Up ∂D j−1[V ]∂α ∂v j−∆t ∂HUpBy the definition of H Up we have∂v j−1if i = j − 1+ ∂HUp∂β]∂D j[V ]∂v jif i = j∂D j[V ]∂β ∂v ⎪⎩j+1if i = j + 10 otherwise.∂H Up∂α (α, β) = {H ′ (α) ≥ 0 if α ≥ α 0 ,0 otherwise.∂H Up∂β (α, β) = {H ′ (β) ≤ 0 if β ≤ α 0 ,0 otherwise.(5.78)(5.79)(5.80)Looking at the signs of the various terms, it is apparent that∂∂v iS Upj (∆; V ) ≥ 0 (i ≠ j), (5.81)whereas, for i = j, we have∂H Up ∂D j−1 [V ]∣ ∂α ∂v j+ ∂HUp∂β∣∂D j [V ] ∣∣∣≤ 2M H ′(L V )∂v j ∆x(5.82)where L V denotes the Lipschitz constant of the sequence V . Therefore, using (5.82)in (5.78), we obtain that the scheme is monotone if∆t∆x ≤ 12M H ′(L V ) . (5.83)Note that, in contrast to the linear case, this condition is more stringent then theCFL condition.

✐✐5.2. Treating the convex HJ equation 131Convergence estimatesFinally, we give the convergence result, which follows from consistency (5.75), monotonicity(5.83), and theorem 4.20.Theorem 5.7. Let H satisfy the basic assumptions, u 0 ∈ W 1,∞ , u be the solution of(5.70) with L as its Lipschitz constant and v n j be defined by (5.73) with v0 j = u 0(x j ).Then, for any j ∈ Z and n ∈ [1, T/∆t],∣∣v n j − u(x j , t n ) ∣ ∣ ≤ C∆t 1/2 (5.84)as ∆t → 0, with 2M H ′(L + 1)∆t ≤ ∆x.5.2.2 Central discretizationIn treating the Lax–Friedrichs scheme, we will follow again the guidelines of [CL84].Rather than using more general forms of the scheme, we will restrict here to theparticular form that directly generalizes the linear case.Construction of the schemeThe simplest way to recast Lax–Friedrichs scheme for the HJ equation is to defineit in the formv n+1j = vn j−1 + vn j+1− ∆tH ( D c2j[V n ] ) , (5.85)where Dj c[V n ] is the centered difference at x j defined byD c j[V n ] = vn j+1 − vn j−12∆x= D j−1[V n ] + D j [V n ]. (5.86)2This definition of the LF scheme completely parallels the linear case, and is alsosuitable to be treated in the framework of The Crandall–Lions theorem. In fact,once recalled thatv n j−1 + vn j+12(5.85) can be written in the differenced formby setting= v n j + ∆x2 (D j[V n ] − D j−1 [V n ]) ,v n+1j = v n j − ∆tH LF (D j−1 [V n ], D j [V n ])( ) α + βH LF (α, β) = H − ∆x (β − α).2 2∆tNote that, as for the advection equation, no special care is necessary to determinethe direction of propagation for the solution (that is, to compare α and β with α 0 ),since the stencil is symmetric.ConsistencyThe Lax–Friedrichs scheme (5.85) satisfies condition (4.51), and in fact( ) a + aH LF (a, a) = H = H(a). (5.87)2Consistency is therefore satisfied.

✐✐132 Chapter 5. First-order approximation schemesStabilityWe examine again the issues of CFL condition and monotonicity, which in this casegive the same restriction on the discretization steps.CFL condition Taking into account that the maximum speed of propagation isM H ′(L), the CFL condition readsM H ′(L V )∆t∆x≤ 1 (5.88)as for the upwind scheme. In this case, this condition is necessary and sufficient,since it also ensures monotonicity (as we will soon show).Monotonicity In examining monotonicity, it is convenient to refer to the LFscheme in the form (5.85). Clearly, the j–th component SjLF (∆; V ) depends onlyon the values v j±1 , so thatOn the other hand, if i = j ± 1, we have∂∂v iS LFj (∆; V ) = 0 (i ≠ j ± 1)∂∂v j±1S LFj (∆; V ) = 1 2 − ∆tH′ ( D c j[V ] ) ∂Dc j [V ]∂v j±1=where we have used the fact that= 1 2 ∓ ∆t2∆x H′ ( D c j[V ] ) ,∂D c j [V ]∂v j±1= ± 12∆x .Therefore, if L V is the Lipschitz constant of the sequence V , the scheme is monotoneprovided∆t∆x ≤ 1M H ′(L V ) . (5.89)Convergence estimatesAgain, the convergence result is obtained from consistency (5.87), monotonicity(5.89), applying theorem 4.20.Theorem 5.8. Let H satisfy the basic assumptions, u 0 ∈ W 1,∞ , u be the solution of(5.70) with L as its Lipschitz constant and v n j be defined by (5.85) with v0 j = u 0(x j ).Then, for any j ∈ Z and n ∈ [1, T/∆t],as ∆t → 0, with M H ′(L + 1)∆t ≤ ∆x.∣ vnj − u(x j , t n ) ∣ ∣ ≤ C∆t1/2(5.90)

✐✐5.2. Treating the convex HJ equation 1335.2.3 Semi–Lagrangian discretizationUnless for replacing the formula of characteristics (1.6)–(1.7) with a suitable generalization,the extension of the Courant–Isaacson–Rees approach to HJ equations isrelatively straightforward. In this chapter we analyse the monotone version of theSL scheme, that is, the version obtained with P 1 interpolation.Construction of the schemeSemi–Lagrangian discretization of the HJ equation follows the same steps seen forthe advection equation: namely, what is really discretized is the representationformula for the solution. In the case of convex HJ equations, the formula underconsideration is the Hopf–Lax formula (2.30). Once rewritten in a single spacedimension, and at a point (x j , t n+1 ) of the space–time grid, it reads:u(x j , t + ∆t) = mina∈R [∆tH∗ (a) + u(x j − a∆t, t)] == ∆tH ∗ (ā j ) + u(x j − ā j ∆t, t) (5.91)(note that, now and then, we might find it useful to introduce the explicit notationā j to denote the minimizer at the node x j in (5.91)). In the special case of (5.70),characteristics are straight lines, so that no special care should be taken about theaccuracy of time discretization (more complex situations will be considered at theend of this section). It is still necessary, however, to replace the value u(x j − a∆t, t)by a space reconstruction. At this stage, and in parallel with the CIR scheme, thiswill be chosen to be the P 1 interpolation I 1 . Once set t = t n , the resulting schemeis therefore:{vn+1j= minα∈R [∆tH∗ (α) + I 1 [V n ](x j − α∆t)]v 0 j = u 0(x j ).(5.92)Since the SL scheme is not in differenced form, the convergence analysis will becarried out in the framework of respectively Barles–Souganidis and Lin–Tadmortheorems.ConsistencyIn order to develop a twofold convergence theory, consistency will be proved separatelyin two different frameworks. In the first, we use the usual notion (which issuitable for the use of Barles–Souganidis theorem), which treats consistency as apointwise concept. In the second, we adopt the L 1 notion of consistency used inthe Lin–Tadmor theory, in the particular form suitable for Godunov-type schemes.Pointwise theory As it has been done for the CIR scheme, the scheme will becompared with the representation formula for the solution, (2.30) in this case. WedefineSjSL (∆; V ) = minα∈R [∆tH∗ (α) + I 1 [V ](x j − α∆t)] == ∆tH ∗ (ᾱ j ) + I 1 [V ](x j − ᾱ j ∆t). (5.93)where ᾱ j denotes the minimizer at the node x j in (5.92).

✐✐134 Chapter 5. First-order approximation schemesLet now u be a smooth solution of (5.70). First, we recall that for any smoothu the error estimate (5.43) is satisfied:‖u − I 1 [U]‖ ∞≤ C∆x 2 .Writing now u(x j , t + ∆t) by means of (5.91), we give a first unilateral estimate onthe left-hand term of (5.94) asL SLj (∆; t, U(t)) = 1 [u(xj , t + ∆t) − Sj SL (∆; t, U(t)) ] =∆t= 1 ∆t [∆tH∗ (ā j ) − u(x j − ā j ∆t, t)−− ∆tH ∗ (ᾱ j ) + I 1 [U(t)](x j − ᾱ j ∆t)] ≤≤ 1 ∆t [∆tH∗ (ᾱ j ) − u(x j − ᾱ j ∆t, t)−− ∆tH ∗ (ᾱ j ) + I 1 [U(t)](x j − ᾱ j ∆t)] ≤≤ 1 ∆t ‖u(t) − I 1[U(t)]‖ ∞≤ C ∆x2∆tNote that the inequality follows from using ᾱ j instead of the exact minimizer ā j in(5.91). By reversing the roles of ᾱ j and ā j we get the opposite inequality,−L SLj (∆; t, U(t)) = 1 ∆t[u(xj , t + ∆t) − SjSL (∆; t, U(t)) ] ≤ C ∆x2∆t ,and therefore,∣∣L SLj (∆; t, U(t)) ∣ ≤ C ∆x2∆t . (5.94)Note that this estimate would suggest that the scheme achieves its best result whengoing to the final time in a single time step. In practice, this advantage is cutdown by some side effects of very large time steps. First, in situation in whichcharacteristics are not straight lines, errors in characteristics tracking should betaken into consideration (see the consistency estimate for the linear case). Second,it might be useful to approximate the solution at intermediate times as well. Third,large time steps enlarge the numerical domain of dependence, thus requiring a morecareful (and computationally more complex) procedure for the global minimumsearch. In respect of this latter argument, it should be noted that in order to obtainany computational advantage from the use of large time steps, it is necessary thatthe complexity of the minimum search would have a weak dependence on the sizeof the set in which this search is performed.L 1 theory Before computing the consistency error in the sense (4.73) of Lin andTadmor, we recall from [LT01] a result which simplifies the computation wheneverthe scheme is in Godunov form – which definitely occurs for SL schemes. For ourpurposes, we will consider a scheme to be in Godunov form when it is based on therepeated application of two operators:• An exact evolution operator E(·) which make the numerical solution evolve(exactly) on a single time step, so that starting from v ∆ (t n ), we define the

✐✐5.2. Treating the convex HJ equation 135numerical solution on (t n , t n+1 ) asv ∆ (t) = E(t − t n )v ∆ (t n ), t n < t < t n+1 ; (5.95)• An operator of projection P ∆x which defines v ∆ (t n+1 ) by projecting the approximationdefined above (computed at the end of the time step) on thespecific space discretization:v ∆ (t n+1 ) = P ∆x v ∆ (t − n+1 ). (5.96)In our case, this operator can be defined as the P 1 interpolation of v ∆ :P ∆x v ∆ (t − n+1 ) = I [1 V ∆ (t − n+1 )] .In the class of Godunov schemes, the local truncation error (4.73) can beexpressed by means of the projection error as follows.Theorem 5.9. [LT01] Let the approximation v ∆ be defined by (5.95)–(5.96). Then,the norm of the consistency error (4.73) may be bounded as:‖F ‖ L 1 (R d ×[0,T ]) ≤ T ∆tmax ∥ v ∆ (t −0

✐✐136 Chapter 5. First-order approximation schemes∫ xj+1I 1 [V ](x) = v(x j ) + x − x jv ′ (ξ)dξ =∆x x j∫ [xj+1x − x j= v(x j ) +v ′ (x j ) +x j∆x∫ ξx jv ′′ (σ)dσ(we denote by 1 [xj,x] the characteristic function of the interval [x j , x] and recall thatthe derivative of the P 1 –interpolate is the integral mean value of v ′ on [x j , x j+1 ]).We can therefore express the interpolation error as|v(x) − I 1 [V ](x)| =∣ v′ (x j )+∫ xj+1(1 [xj,x](ξ) − x − x jx j∆x) ∫ ξ∫ xj+1(1 [xj,x](ξ) − x − x jx j∆x])dξ +dξv ′′ (σ)dσdξx j∣ . (5.99)Note that, in (5.99), the first integral is identically zero, and hence, giving a boundon the second integral by Hölder’s inequality,∫ xj+1∣ ∣∣∣|v(x) − I 1 [V ](x)| ≤ 1 [xj,x](ξ) − x − x ∫jξ∆x ∣ dξ · sup |v ′′ (σ)|dσ ≤ξ x j≤ ∆xx j∫ xj+1x j|v ′′ (σ)|dσ, (5.100)where we have taken into account that the first term is the integral of a function takingvalues in [0, 1], and that the second is maximized setting ξ = x j+1 . Integrating(5.100) on the interval [x j , x j+1 ], we have∫ xj+1x j∫ xj+1|v(x) − I 1 [V ](x)|dx ≤ ∆x 2 |v ′′ (σ)|dσ,and last, summing over all elementary intervals, we obtainx j‖v − I 1 [V ]‖ L1 (R) ≤ ∆x 2 ‖v ′′ ‖ L1 (R). (5.101)Finally, in order to apply Theorem 5.9, we must recall that neither the evolutionoperator, nor the P 1 interpolation expand the L 1 norm of the second derivative, asit can be easily seen by interpreting the solution as the integral of the solution ofthe associated conservation law (2.46). We have thereforeStability‖F ‖ L1 (R×[0,T ]) ≤ T ‖u ′′ ∆x 20‖ L1 (R)∆t . (5.102)Along with some remarks on the CFL condition, we will address here the twoconcepts of nonlinear stability used in the two different convergence theories, namelymonotonicity and uniform semiconcavity.CFL condition Looking at the basic definition (5.92) of the scheme, one could inferthat, since α is allowed to vary over the whole of R, then the numerical domainof dependence also coincides with R and the CFL condition is always satisfied. Inpractice, whenever the Lipschitz constant of u 0 (hence, of u(t) for positive t) isknown, it is possible to restrict the search for a minimum to a smaller subset of R,still without violating the CFL condition.

✐✐5.2. Treating the convex HJ equation 137Monotonicity First, note that the SL scheme is invariant for the addition of constantssince, for a Lagrange interpolation of any order, I[V + c](x) ≡ I[V ](x) + c.To check that the SL scheme is monotone, consider two sequences V and Wsuch that V − W ≥ 0 componentwise. We have:SjSL (∆; V ) = minα∈R [∆tH∗ (α) + I 1 [V ](x j − α∆t)] == ∆tH ∗ (ᾱ j ) + I 1 [V ](x j − ᾱ j ∆t),SjSL (∆; W ) = minα∈R [∆tH∗ (α) + I 1 [W ](x j − α∆t)] == ∆tH ∗ (˜α j ) + I 1 [W ](x j − ˜α j ∆t) ≤≤ ∆tH ∗ (ᾱ j ) + I 1 [W ](x j − ᾱ j ∆t)(as before, the last inequality is obtained by replacing the exact minimizer for W ,˜α j , with ᾱ j ). Therefore,S SLj(∆; V ) − S SL (∆; W ) ≥ I 1 [V ](x j − ᾱ j ∆t) − I 1 [W ](x j − ᾱ j ∆t),jand since P 1 interpolation is monotone itself (that is, I 1 [V ] − I 1 [W ] ≥ 0), we getSjSL (∆; V ) − Sj SL (∆; W ) ≥ 0. (5.103)Uniform discrete semiconcavity Assume that at the n-th step the discrete semiconcavityassumptionvr−h n − 2vr n + vr+h n ≤ C∆x 2 (5.104)holds for the discrete solution at any node x r with an increment ±h∆x (h being aninteger). Then, using the notation of (5.93), at the (n + 1)–th step we havev n+1j−h − 2vn+1 j + v n+1j+h = ∆tH∗ (ᾱ j−h ) + I 1 [V n ](x j−h + ᾱ j−h ∆t) −−2∆tH ∗ (ᾱ j ) − 2I 1 [V n ](x j + ᾱ j ∆t)] ++∆tH ∗ (ᾱ j+h ) + I 1 [V n ](x j+h + ᾱ j+h ∆t) ≤≤ ∆tH ∗ (ᾱ j ) + I 1 [V n ](x j−h + ᾱ j ∆t) −−2∆tH ∗ (ᾱ j ) − 2I 1 [V n ](x j + ᾱ j ∆t) ++∆tH ∗ (ᾱ j ) + I 1 [V n ](x j+h + ᾱ j ∆t)where we have bounded the sum from above by replacing the minimizers for thenodes x j±h , i.e., ᾱ j±h , with the minimizer for the node x j . Now, for I 1 [V n ](x j +ᾱ j ∆t) = ∑ i vn i ψ[1] i (x j + ᾱ j ∆t), by periodicity of the grid we can setψ [1]i (x j + ᾱ j ∆t) = ψ [1]i±h (x j±h + ᾱ j ∆t) = β iso that I 1 [V n ](x j±h + ᾱ j ∆t) = ∑ i β iv n i±h . Recalling that the P 1 reconstructionconsists of a convex combination of the values in neighbouring nodes, we have thatβ i ≥ 0 and ∑ i β i ≡ 1. By we obtain thereforev n+1j−h − 2vn+1 j + v n+1j+h ≤ ∑ iβ i v n i−h − 2 ∑ iβ i v n i + ∑ iβ i v n i+h == ∑ i≤ ∑ iβ i (v n i−h − 2v n i + v n i+h) ≤β i C∆x 2 = C∆x 2 , (5.105)

✐✐138 Chapter 5. First-order approximation schemesand the solution satisfies the same semiconcavity estimate at the (n + 1)–th step.ConvergenceIn view of the consistency estimate (5.94) and the monotonicity result (5.103), convergenceof the first-order SL scheme follows from the Barles–Souganidis theorem.Theorem 5.10. Let H satisfy the basic assumptions, u 0 ∈ W 1,∞ , u be the solutionof (5.70) and vjn be defined by (5.92) with v0 j = u 0(x j ). Then, for any j ∈ Z andn ∈ [1, T/∆t], ∣ vj n − u(x j , t n ) ∣ → 0 (5.106)locally uniformly, as ∆t → 0, with ∆x = o ( ∆t 1/2) .The parallel result, obtained by means of the Lin–Tadmor convergence theorem,uses the consistency estimate (5.102) and the uniform semiconcavity result(5.105), and may be stated as follows.Theorem 5.11. Let H satisfy the basic assumptions, u 0 ∈ W 1,∞ , u be the solutionof (5.70) and v n j be defined by (5.92) with v0 j = u 0(x j ). Assume moreover that u 0is semi-concave and compactly supported. Then, for any j ∈ Z and n ∈ [1, T/∆t],‖u(t n ) − I 1 [V n ]‖ L1 (R) ≤ C ∆x2∆t(5.107)as ∆t → 0, with ∆x = o ( ∆t 1/2) .5.2.4 Multiple space dimensionsAs it has been done for the advection equation, we briefly turn to the n–dimensionalproblem:u t + H(u x1 , . . . , u xd ) = 0 R d × [0, T ]. (5.108)Most of the work of extending one-dimensional schemes for HJ equations to themultidimensional case follows the same principles of the linear case. In particular,schemes in differenced form have the general structure (4.50), so that for examplethe two-dimensional version of the Lax–Friedrichs scheme readsv n+1j 1,j 2= 1 (vn4 j1−1,j 2+ vj n 1+1,j 2+ vj n 1,j 2−1 + vj n )1,j 2+1 +( vnj1+1,j+∆tH2− vj n 1−1,j 2, vn j − )1,j 2+1 vn j 1,j 2−1,∆x 1∆x 2and both consistency and monotonicity may be proved by the very same argumentsused for the one-dimensional case. Also, the d-dimensional form of the SL scheme,v n+1j= minα∈R d [∆tH ∗ (α) + I 1 [V n ](x j − α∆t)]retains the same formal structure of the one-dimensional form, although (as it hasbeen noticed for the linear scheme) reconstruction and minimization are now performedin R d . Again, the proofs of consistency (which is obtained by comparisonwith the Hopf–Lax formula) and monotonicity follow the same criteria used in asingle space dimension.

✐✐5.2. Treating the convex HJ equation 1395.2.5 Boundary conditionsAs we have seen in Chapter 2, boundary conditions for HJ equations must beinterpreted in a weak sense, with the only exception of periodic conditions which canbe numerically implemented as it has been sketched for linear problems. Concerningboundary conditions of other kinds, we will limit ourselves here to the basic ideasfor their implementation in SL schemes. In this case, the typical technique has beenoutlined in the treatment of Dirichlet conditions for the advection equation, whichrequire a variable step algorithm to bring the foot of characteristics precisely on theboundary. In addition, it might happen that boundary conditions are applied atnodes belonging to a neighborhood of the boundary instead that just on the boundarynodes (as it is usual in classical finite difference schemes). This point will be usefulin treating Neumann condition.We consider three different boundary conditions: Dirichlet, Neumann andState Constraint.Dirichlet boundary conditions The boundary operator for Dirichlet conditions isB(x, u, Dϕ(x)) = u(x) − b(x), where b is the value imposed on the boundary ∂Ω.Characteristics ending outside Ω should be brought to the boundary following thesame idea used in (5.65). As a result, (5.92) is modified asv n+1j = min [∆tH ∗ (α) + I 1 [V n ](x j − ατ)] (5.109)α∈R dτ≤∆tx j−ατ∈Ωwhere the value I 1 [V n ](x j − ατ) should be understood asI 1 [V n ](x j − ατ) = b(x j − ατ)if x j − ατ ∈ ∂Ω. The minimization in (5.109) should be interpreted as follows. Theminimum is searched for with τ = ∆t. If the minimum is found at an internal point(i.e., for x j − ᾱ j ∆t ∈ Ω), then nothing else is necessary. If no minimum point isfound in the interior of Ω, then among all boundary points which can be writtenas x j − ᾱ j τ, it is necessary to select the one which achieves the minimum withrespect to α ∈ R d and τ ≤ ∆t. In the chapter on Dynamic Programming, we willre-interpret this boundary condition as a stopping cost.Neumann boundary conditionsHere, the boundary operator isB(x, u, Du) = ∂u (x) − m(x),∂νwhere m is a given function. In order to assign a value to points outside (but closeto) Ω, we can use the directional derivative on the boundary to writeu(x + δν(x)) = u(x) + δm(x), x ∈ ∂Ω. (5.110)Then, Neumann boundary conditions can be enforced by using directly (5.110) toreplace the value I 1 [V n ](x j − α∆t) whenever x j − α∆t /∈ Ω. A different way is toextend the grid outside the domain Ω with the introduction of a neighborhood inwhich the values are assigned according to (5.110) and reconstructed as usual byinterpolation in (5.92).

✐✐140 Chapter 5. First-order approximation schemesFor example, if m(x) ≡ 0 and Ω is a rectangle of R 2 , then we just need to addan extra “frame” of nodes surrounding the boundary at a distance δ outside Ω, thecorresponding values being defined asu(x + δν(x)) = u(x),x ∈ ∂Ω.Clearly, the points x j − α∆t must be kept within this enlarged domain, either by asuitable choice of δ, or by a variable step technique.State constraints boundary conditions This boundary condition is the easiest toimplement. In fact, we recall that what is needed is just to extend the solution uoutside Ω by settingu(x) = C, x ∈ R d \ Ωwhere C is an upper bound for the L ∞ (Ω) norm of the solution. In the numericalimplementation, in order to preserve the regularity of the function to be minimizedin (5.92), it could be necessary to implement again a variable step algorithm, with Cas a Dirichlet boundary datum. Note that, in a Dynamic Programming framework,this “large” value could be interpreted as a penalization (made through the stoppingcost) of trajectories which violate the state constraints.5.3 ExamplesWe present a new set of simple numerical examples, using the one-dimensionalmodel problem{u t (x, t) + 1 2 |u x(x, t)| 2 = 0 (x, t) ∈ (0, 1) × (0, T )(5.111)u(x, 0) = u 0 (x)with T = 0.05 and two different initial conditions u 0 with bounded support. Thefirst is a Lipschitz continuous function (the same used for the tests of subsection5.1.6):u 0 (x) = max(1 − 16(x − 0.25) 2 , 0), (5.112)whereas the second, obtained by a simple change of sign, is also semiconcave:u 0 (x) = − max(1 − 16(x − 0.25) 2 , 0). (5.113)Using the initial condition (5.112), the solution eventually develops a singularitywith nonempty superdifferential. After the onset of the singularity, the exact solutionreads{(|x−12|− 1 4) 2u(x, t) = 2tif 1 4 ≤ x ≤ 3 40 elseOn the other hand, using the initial condition (5.113), the solution has the expression(∣ ∣x −1∣ 2 )2u(x, t) = min2t + 1 − 1, 0 .16Again, the test is performed with Upwind, Lax–Friedrichs and Semi-Lagrangianschemes. In this case, the refinement has been carried out with ∆t = ∆x/40 forUpwind scheme, ∆t = ∆x/20 for Lax–Friedrichs scheme and ∆t = 0.01 (fixed)

✐✐5.4. Stationary problems 141W 1,∞ initial conditionn n Upwind LF SL25 1.13 · 10 −1 2.84 · 10 −1 8.82 · 10 −250 1.01 · 10 −1 2.51 · 10 −1 3.53 · 10 −2100 6.62 · 10 −2 1.89 · 10 −1 1.81 · 10 −2200 4.08 · 10 −2 1.27 · 10 −1 8.26 · 10 −3400 2.42 · 10 −2 8.0 · 10 −2 3.94 · 10 −3rate 0.56 0.46 1.12Semiconcave initial conditionn n Upwind LF SL25 6.42 · 10 −2 3.64 · 10 −1 2.11 · 10 −250 3.58 · 10 −2 1.97 · 10 −1 5.02 · 10 −3100 1.92 · 10 −2 9.76 · 10 −2 1.24 · 10 −3200 9.97 · 10 −3 4.83 · 10 −2 3.07 · 10 −4400 5.08 · 10 −3 2.4 · 10 −2 7.63 · 10 −5rate 0.91 0.98 2.03Table 5.3. Errors in the ∞-norm for problem (5.111), Upwind, Lax–Friedrichs and Semi-Lagrangian schemes.for Semi-Lagrangian scheme. Table 5.3 shows numerical errors in the ∞-norm andFigure 5.8 compares exact with numerical solutions for the three schemes in bothexamples.A first clear outcome of this example is that, although both initial conditionsare Lipschitz continuous, having a semiconcave initial condition (and hence,a uniformly semiconcave solution) makes a difference. In the first test the schemesbasically respect the theory, by keeping a convergence rate of about 1/2 for Upwindand Lax–Friedrichs schemes, of about 1 for Semi-Lagrangian scheme (note that thisis the order of truncation error for a Lipschitz solution in the ∞-norm, once thetime step is kept constant). On the other hand, in the second test convergence ratesare almost doubled. A heuristic explanation of this effect is provided by Remark2.25: the Hamiltonian and its conjugate being smooth and coercive, the foot ofa characteristic must be a point of differentiability for u. Roughly speaking, thismeans that a semiconcave solution always propagates from regular points towardssingularities, and this reduces numerical errors.Note that, in spite of a similar convergence rate, LF scheme gives a poor approximationin terms of absolute accuracy, even with respect to Upwind scheme. SLscheme has the best performances in both convergence rate and absolute accuracy,still being monotone.5.4 Stationary problemsIn this section, we will sketch how the theory introduced so far can be adapted tostationary problems, respectively of linear and Hamilton–Jacobi type. As a generalstrategy, stationary equations will be regarded as limit states of evolutive equations,

✐✐142 Chapter 5. First-order approximation schemesUpwindLax–FriedrichsSemi-LagrangianFigure 5.8. Numerical results for problem (5.111) with Lipschitz (left) andsemiconcave (right) initial condition, obtained via Upwind (upper), Lax–Friedrichs(center) and Semi-Lagrangian (lower) schemes, 200 nodes.and as a consequence it will be natural to consider schemes in the time-marchingform.5.4.1 The linear caseWhen posed on the whole of R, a general form of the stationary linear advectionequation isu(x) + f(x)u x (x) = g(x) x ∈ R, (5.114)

✐✐5.4. Stationary problems 143whose solution u(x) can be considered as the limit for t → ∞ of a solution u(x, t)of the equationu t (x, t) + u(x, t) + f(x)u x (x, t) = g(x) x ∈ R, (5.115)regardless of the initial condition used. Note that equations having a differentcoefficient of the zeroth order term u(x) may be brought to this form by a rescaling.We will proceed to write the various schemes in the time-marching form (4.19)by applying them to (5.115). Note that the extension to more general situations(multiple space dimensions, boundary conditions) can be carried out in a fairlystraightforward way by following the guidelines given for evolutive problems.Upwind discretizationFollowing the general strategy outlined in the first part of the chapter, (5.115) wouldbe discretized, for f(x j ) > 0, asv n+1jand for f(x j , t n ) < 0, as− v n j∆t+ v n j + f(x j ) vn j − vn j−1∆x= g(x j ),v (k+1)j =⎪⎨v n+1j− v n j∆t⎪⎩ (1 − ∆t)v (k)+ v n j + f(x j ) vn j+1 − vn j∆x= g(x j ).Once replaced the index of the time step by the index of iteration, the form of thescheme for f(x) changing sign is therefore⎧(1 − ∆t)v (k)j − f(x j ) ∆t [ ]v (k)j − v (k)j−1 + ∆tg(x j ) if f(x j ) > 0∆x(1 − ∆t)v (k)j + ∆tg(x j ) if f(x j ) = 0j− f(x j ) ∆t [ ]v (k)j+1∆x− v(k) j + ∆tg(x j ) if f(x j ) < 0.(5.116)Here and in the whole section, since we look for the limit solution, the choice of theinitial vector V (0) is irrelevant.Convergence Once put the scheme in the form (4.20), it is easy to check that,under the slightly more restrictive CFL condition‖f‖ ∞ ∆t∆x< 1 − ∆t (5.117)the scheme satisfies assumption (4.21), and more precisely ‖B Up (∆)‖ ∞ = 1 − ∆t.There also follows (see Subsection 4.2.5) that it satisfies the consistency estimate(5.10), and converges by Theorem 4.12. We have therefore:Theorem 5.12. Let f, g ∈ W 1,∞ , u be the solution of (5.114) and v j = lim k v (k)j ,with v (k)j defined by (5.116). Then, for any j ∈ Z,|v j − u(x j )| → 0 (5.118)

✐✐144 Chapter 5. First-order approximation schemesas ∆t, ∆x → 0, with the CFL condition (5.117).Moreover, if u has bounded second derivative, then‖V − U‖ ∞≤ C∆x. (5.119)Central discretizationFor simplicity, Lax–Friedrichs scheme will be adapted to equation (5.115) using oncemore the approximationv(x j ) = v(x j−1) + v(x j+1 )2+ O(∆x 2 )in treating the zeroth-order term. With this choice, the time-marching form of theLF scheme isv (k+1)j= (1 − ∆t) v(k) j−1 + v(k) j+1− f(x j ) ∆t []v (k)j+122∆x− v(k) j−1 + ∆tg(x j ). (5.120)Convergence Comparing (5.120) with (4.20), it is easily seen that the LF schemesatisfies the stability condition ‖B LF (∆)‖ ∞ = 1 − ∆t under the CFL constraint(5.117). Moreover, as for the upwind scheme, LF scheme in this form retains theconsistency estimate of the evolutive scheme, i.e. (5.24). Therefore, applying Theorem4.12, we obtain the following convergence result:Theorem 5.13. Let f, g ∈ W 1,∞ , u be the solution of (5.114) and v j = lim k v (k)j ,with v (k)j defined by (5.120). Then, for any j ∈ Z,|v j − u(x j )| → 0 (5.121)as ∆t, ∆x → 0, with ∆x = o ( ∆t 1/2) and the CFL condition (5.117).Moreover, if u has bounded second derivative and ∆x = c∆t for some constant c,then‖V − U‖ ∞≤ C∆x. (5.122)Semi-Lagrangian discretizationTo write the time-marching version of the CIR scheme, we start from the representationformula (5.36) which is changed into its more general versionu(x, t) =∫ tt−∆te s−t g(y(x, t; s))ds + e −∆t u(y(x, t; t − ∆t), t − ∆t). (5.123)Here, the ODE satisfied by characteristics is driven by a vector field f(x) whichdoes not depend on t. We can therefore set conventionally t = 0 and omit theinitial time in the notation of y, thus obtaining the representation formulau(x, t) ==∫ 0−∆t∫ ∆t0e s g(y(x; s))ds + e −∆t u(y(x; −∆t), t − ∆t) =e −s g(y(x; −s))ds + e −∆t u(y(x; −∆t), t − ∆t).

✐✐5.4. Stationary problems 145In order to treat this case, the augmented system (5.52) is modified in the form(ẏ(x; ) ( )s) f(y(x; s))=˙γ(x; s) −e −s (5.124)g(y(x; s))with the initial conditions y(x; 0) = x, γ(x; 0) = 0. Denoting again by X ∆ (x; s) andG ∆ (x; s) the approximations of y(x; s) and γ(x; s), and replacing the time index bythe iteration index, we finally obtain the fixed point formv (k+1)j = G ∆ (x j ; −∆t) + e −∆t I 1[V (k)] (X ∆ (x j ; −∆t) ) . (5.125)Convergence Following the arguments used for the evolutive case, we can checkthat the scheme satisfies the consistency estimate (5.57), and that ‖B CIR (∆)‖ ∞ =e −∆t = 1 − ∆t + O ( ∆t 2) . It is therefore possible to apply Theorem 4.12 to provethe following convergence result:Theorem 5.14. Let f, g ∈ W p,∞ , u be the solution of (5.114) and v j = lim k v (k)with v (k)j defined by (5.125). Assume moreover that (5.55), (5.56) hold. Then, forany j ∈ Z,|v j − u(x j )| → 0 (5.126)as ∆t → 0, ∆x = o ( ∆t 1/2) .Moreover, if u ∈ W s,∞ (R) (s = 1, 2), then‖V − U‖ ∞≤ C(∆t p + ∆xs∆tj ,). (5.127)5.4.2 The nonlinear caseIn adapting the various schemes to stationary HJ equations, we refer to the stationarymodel which in some sense parallels (5.114), that isu(x) + H(u x (x)) = 0 x ∈ R. (5.128)As before, we consider time-marching schemes, either in differenced form or ofSemi-Lagrangian type.Discretization in differenced formIn differenced time-marching schemes, the schemes are applied to the evolutiveequationu t + u + H(u x ) = 0,whose solution converges to a regime state satisfying (5.128). Keeping the schemein its most general form, ad adding the zeroth order term, we havev n+1j− v n j∆twhich is clearly a consistent scheme.iteration index, we obtainv (k+1)j= (1 − ∆t)v (k)j+ v n j + H (D j−p [V n ], . . . , D j+q [V n ]) = 0,Hence, replacing the time index with the+ ∆tH(D j−p[V (k)] [, . . . , D j+q V (k)]) . (5.129)

✐✐146 Chapter 5. First-order approximation schemesrepeating the computations performed for the evolutive case, we can show that boththe upwind and the Lax–Friedrichs scheme satisfy the assumptions of the stationaryversion of Barles–Souganidis theorem (Theorem 4.26), with a contraction coefficientof L S = 1−∆t, provided a suitable upper bound on the Courant number is satisfied.Both schemes are therefore convergent to the viscosity solution of (5.128):Theorem 5.15. Let H satisfy the basic assumptions, u be the solution of (5.128)and v j = lim k v (k)j , with v (k)j defined by (5.129). Assume moreover that the numericalHamiltonian H is consistent and monotone in the sense of Theorem 4.20. Then,for any j ∈ Z,|v j − u(x j )| → 0 (5.130)as ∆ → 0.Semi-Lagrangian discretizationIn the case of Semi-Lagrangian discretization, we start from a generalized form ofthe Hopf–Lax formula, which applies to the solution of (5.128):[(u(x) = min 1 − e−∆t ) H ∗ (a) + e −∆t u(x − a∆t) ] .a∈RA detailed derivation of this representation formula will be shown in the chapterdevoted to Dynamic Programming, whereas for the moment we simply use it toconstruct a SL type discretization. Replacing the value u(x − a∆t) with its P 1numerical reconstruction, we have the scheme (in iterative form):v (k+1)j= mina∈R[ (1− e−∆t ) H ∗ (a) + e −∆t I 1[V (k)] (x − a∆t)]. (5.131)Adapting the arguments used for the evolutive case, it can be shown once morethat the scheme is consistent for ∆x = o ( ∆t 1/2) and monotone for any ∆t/∆xrelationship. We can therefore apply Theorem 4.26 to state the convergence result:Theorem 5.16. Let H satisfy the basic assumptions, u be a bounded and uniformlycontinuous solution of (5.114) and v j = lim k v (k)j , with v (k)j defined by (5.131).Then, for any j ∈ Z,|v j − u(x j )| → 0 (5.132)as ∆t → 0, ∆x = o ( ∆t 1/2) .5.5 Commented referencesThe linear theory for classical monotone difference schemes like Upwind or Lax–Friedrichs is widely treated in any textbook on difference schemes. A relativelyrecent reference on this subject is [Str89]. However, the application of such schemesto Hamilton–Jacobi equations is a much less established topic, and we are notaware of other monographs covering the subject. Unless for some more explicitcomputation, the theory reported here is basically taken from [CL84].The Courant–Isaacson–Rees has been proposed in [CIR52], and although theprocedure of construction outlined in the paper clearly implies interpolation, itsendpoint basically coincides with an upwind scheme. Moreover, the possibility ofworking at large Courant numbers was not recognized by that time. Most authors

✐✐5.5. Commented references 147believe [W59] to be the first work proposing a Semi-Lagrangian technique. In fact,[W59], along with the later paper [Ro81], has originated a considerable streamlineof literature concerning the application of SL techniques to Numerical Weather Predictionand environmental Fluid Dynamics, for which [SC91] represents a classical(although no longer up-to-date) review paper.In a completely independent way, SL schemes have been developed in thecomputational study of plasma physics. To our knowledge, the first paper presentingthis approach is [CK76], whereas further developments may be found in[GBSJFF90], [SBG99], [BM08].Once more independently, the Hopf–Lax representation formula (or its equivalentin Optimal Control, the Dynamic Programming Principle) has been used inorder to construct numerical schemes for HJ equations. This approach has originatedin the 70s with the probabilistic techniques of the so-called Markov chainapproximations, for which a recent review can be found in the monograph [KuD01].After the emergence of the theory of viscosity solutions, a reformulation of thisapproach has been given for deterministic problems. A review on this streamline ofresearch is given in [F97]. We also mention the papers [FF94], in which the couplingbetween a monotone reconstruction and a high-order time discretization has beenfirst proposed, and [FF98, FF02] which analyze the SL scheme for respectively theadvection and the HJ equation in a more numerical framework.

✐✐148 Chapter 5. First-order approximation schemes

✐✐Chapter 6High-order SLapproximation schemesIt is well known that, despite the increasing number of accurate and efficient highorderschemes available in the literature, the related convergence theory is usuallypoorer. This chapter presents the state-of-art of theoretical analysis for high-orderSemi-Lagrangian schemes in the fields of transport and Hamilton–Jacobi equations.The presentation is dedicated to SL schemes built by means of Lagrange, finiteelement and non-oscillatory reconstructions. In its structure, the chapter parallelsthe previous one, unless for the introduction of different and more complex toolsat the level of stability analysis. Also, a subsection of one-dimensional examples isincluded for both the linear and the nonlinear model, and a qualitative study of theinterplay between the two discretization steps is presented.6.1 Semi-Lagrangian schemes for the advectionequationIn turning our attention to high-order Semi-Lagrangian schemes, we get back to thecase of the one-dimensional linear advection equation, which is rewritten here as{u t (x, t) + f(x, t)u x (x, t) = g(x, t) (x, t) ∈ R × [t 0 , T ](6.1)u(x, 0) = u 0 (x) x ∈ R.As it has been done in the previous chapter, we will also examine in detail the casewith constant coefficients, namely,(with c > 0) whenever useful to illustrate the main ideas.6.1.1 Construction of the schemeu t + cv x = 0 (6.2)A first step to obtain a fully high-order discretization of the representation formula(5.32), (5.36) has already been presented concerning the CIR scheme and the possibilityto track characteristics in a more accurate way. This increase in the accuracyof time discretization for the augmented system (5.60) is summarized in Theorem5.6. The remaining step is to replace the P 1 reconstruction used in the CIR schemewith an interpolation of order r, which will be denoted by I r .149

✐✐150 Chapter 6. High-order SL approximation schemesWith this further improvement, the SL scheme can be rewritten as{v n+1j = G ∆ (x j , t n+1 ; t n ) + I r [V n ] ( X ∆ (x j , t n+1 ; t n ) )vj 0 = u 0(x j ).(6.3)Note that, as it has been remarked in the first-order case, this form also applies toa scheme in d space dimensions, as soon as the augmented system (5.52) is set indimension d + 1 and I r is a d-dimensional reconstruction.6.1.2 ConsistencyConsistency analysis for the scheme (6.3) requires the basic estimates (5.55)–(5.56),which read∣ X ∆ (x j , t n+1 ; t n ) − y(x j , t n+1 ; t n ) ∣ = O(∆t p+1 ),∫ tn+1∣ G∆ (x j , t n+1 ; t n ) − g(y(x j , t n+1 ; s), s)ds∣ = O(∆tp+1 ),as well as the corresponding estimate for interpolation:t n‖u − I r [U]‖ ∞= O(∆x r+1 ). (6.4)Besides assuming that f, g ∈ W p,∞ for (5.55), (5.56) to hold, (6.4) requires typicallythat u ∈ W r+1,∞ . Then, retracing the proof of (5.44), (5.57), and incorporatingthe interpolation error (6.4), we obtain the consistency estimate∥ L SL (∆; t, U(t)) ∥ ()≤ C ∆t p + ∆xr+1. (6.5)∆tNote that the scheme ( is)still conditionally consistent, but under a weaker constraint,that is, ∆x = o ∆t 1r+1 .6.1.3 StabilityWhile high-order characteristics tracking does not change the stability propertiesof the CIR scheme, the introduction of a space reconstruction of degree r > 1 does.In this situation, the scheme is no longer monotone, and a completely differentstability analysis must be performed.A short digression: the Lagrange–Galerkin schemeIn this small detour, we introduce a class of schemes which will be useful in the sequelto analyse the stability of SL schemes under Lagrange reconstructions. Similar toSL schemes, Lagrange–Galerkin (LG) schemes are constructed by a discretizationof the representation formula in the form (5.32) or (5.36). Unlike the SL schemes,however, LG schemes perform the space reconstruction by means of a Galerkinprojection.To give a more precise expression, assume for simplicity that g ≡ 0, so thatthe representation formula for the solution would bev(x, t n+1 ) = v(y(x, t n+1 ; t n ), t n ). (6.6)

✐✐6.1. Semi-Lagrangian schemes for the advection equation 151The first step is to write the numerical solution at time t n , v n h , in the base {φ i}:v n h(x) = ∑ iv n i φ i (x). (6.7)Note that we have used for the numerical solution the notation vh n , usual in Galerkinschemes (here, typically h denotes the space discretization step ∆x), and that wehave discriminated the Galerkin basis function φ i from the basis functions ψ i usedfor interpolation. In fact, the Galerkin basis functions have a conceptually differentrole, and in particular need not to be cardinal functions. Rather, they are assumedto be in L 2 , and to be asymptotically dense as ∆x → 0. The steps to turn therepresentation formula (6.6) into a LG scheme are:• replacing the exact solution by its approximation (6.7) and the exact displacementalong characteristics y with its approximation X ∆ ;• multiplying both sides by a test function w h = ∑ i wn i φ i(x);• integrating with respect to the space variable.This amounts to define the scheme by∫∫v n+1h(ξ)w h (ξ)dξ = vhn (X ∆ (ξ, t n+1 ; t n ) ) w h (ξ)dξ, (6.8)RRwhere the equality has to hold for any w h in the space generated by the base {φ i }.Hence, using (6.7) and using as test functions the basis functions φ j , the resultingscheme is set in the form:∫∑∫∑v n+1i φ i (ξ)φ j (ξ)dξ = vi n (φ i X ∆ (ξ, t n+1 ; t n ) ) φ j (ξ)dξRiRiwhere j ranges over the set of admissible indices I. This condition is actuallyenforced as∑∫v n+1i φ i (ξ)φ j (ξ)dξ = ∑ ∫v n (i φ i X ∆ (ξ, t n+1 ; t n ) ) φ j (ξ)dξ. (6.9)iR i RNote now that the LG scheme, as it has been defined, ( is stable in L 2 . Forexample, in the constant coefficient case, we have that vhn X ∆ (ξ, t n+1 ; t n ) ) =vh n (ξ − c∆t)) is a pure translation, so that ∥ ( vnhX ∆ (·, t n+1 ; t n ) )∥ ∥2= ‖vh n(·))‖ 2 .Therefore, using w h = v n+1has a test function in (6.8), and applying Hölder’s inequality,we get∫∥ vn+1∥ 2 h 2 = v n (h X ∆ (ξ, t n+1 ; t n ) ) v n+1h(ξ)dξ ≤R∥ ∥≤ ‖vh‖ n ∥v n+12hand this shows that the scheme is stable in the L 2 norm. More in general, the LGscheme is stable whenever the approximate evolution operator E ∆ defined byE ∆ (t − t n )vh(x) n = vhn (X ∆ (x, t; t n ) )∥2

✐✐152 Chapter 6. High-order SL approximation schemessatisfies for t − t n → 0 + the bound:∥ E ∆ (t − t n ) ∥ ∥ ≤ 1 + C(t − tn ).in which the left-hand side is understood as the norm of an operator mapping L 2into L 2 .It is worth to point out that, in fact, the definition (6.8) of LG schemesdoes not allow in general for an exact implementation – this happens because thebasis function are deformed by the approximate advection X ∆ , and the integralsappearing at the right-hand side of (6.8) cannot be exactly evaluated, unless forconstant advection speed. Thus, the real implementation of LG schemes requiressome specific techniques, like approximate integration through quadrature formulaeor area-weighting (we will get back to this latter technique when treating variablecoefficient equations). Rather then giving details on the implementation of LGschemes, our interest here is to use the stability of their exact version (6.8) toprove stability of SL schemes, via a result of equivalence between the two classes ofschemes. This tool will be the used to treat the case of Lagrange reconstructions.The constant coefficient caseFor the constant coefficient case, the SL scheme (6.3) takes the form{v n+1j = I[V n ](x j − c∆t) = ∑ i vn i ψ i(x j − c∆t)vj 0 = u 0(x j )(6.10)in which we have expressed the interpolation in terms of the basis functions. Dependingon the strategy of reconstruction used, the scheme would require a differenttheory. Here, we will address two cases: the first is the case of odd order Lagrangereconstruction, for which the theory is essentially complete, the second is the caseof finite element reconstructions, for which a partial Von Neumann type analysiswill be presented.Lagrange reconstructions In this situation, the reconstruction is invariant bytranslation and Von Neumann analysis (by means of Fourier methods) gives a necessaryand sufficient stability condition. The Von Neumann condition (4.47) readsin this case as∣ ∑∣∣∣∣|ρ(ω)| =ψ∣ l (x j − c∆t) e iω(l−j) ≤ 1 + C∆t.lAlthough an explicit proof of this inequality has been given in [BM08], such aproof is very technical and an easier way consists in showing that the SL scheme isequivalent to a Lagrange–Galerkin scheme. We recast therefore (6.9), for the caseunder consideration, as∑∫v n+1i φ i (ξ)φ j (ξ)dξ = ∑ ∫vin φ i (ξ − c∆t)φ j (ξ)dξ. (6.11)iR i RThe equivalence between (6.10) and (6.11) is stated by the following

✐✐6.1. Semi-Lagrangian schemes for the advection equation 153Theorem 6.1. Let the functions ψ i be defined by( ) ξψ i (ξ) = ψ∆x − i , (6.12)for some reference function ψ such thatψ ∈ W 2,1 (R) (6.13)ψ(y){= ψ(−y) (6.14)ψ(i) =1 if i = 00 if i ∈ Z, i ≠ 0.(6.15)Then, there exists a basis {φ i } such that the SL scheme (6.10) is equivalent to theLG scheme (6.11) if and only the function ψ(y) has a real nonnegative Fouriertransform:ˆψ(ω) ≥ 0. (6.16)Proof. We look for a set of basis functions for the LG scheme in the formφ i (ξ) = √ 1 ( ) ξφ∆x ∆x − i(6.17)for some reference function φ to be determined. Comparing (6.10) and (6.11), weobtain the set of conditions to be satisfied:∫φ i (ξ)φ j (ξ)dξ = δ ij (6.18)∫RRφ i (ξ − c∆t)φ j (ξ)dξ = ψ i (x j − c∆t). (6.19)Note that, by assumption (6.15), condition (6.18) is in fact included in (6.19), asit can be seen setting ∆t = 0. Using the definitions (6.12), (6.17) for ψ k and φ k ,(6.19) can be rewritten as1∆x∫R( ξ − c∆tφ∆x) ( ) ( )ξ− i φ∆x − j xj − c∆tdξ = ψ− i∆xthat is, after defining η = ξ/∆x − j and λ = c∆t/∆x:∫φ(η − λ + j − i)φ(η)dη = ψ(−λ + j − i).R(6.20)This ultimately amounts to find a function φ such that∫φ(η + y)φ(η)dη = ψ(y). (6.21)RThe left-hand side of (6.21) is the autocorrelation integral (see [P77]) of the unknownfunction φ. Working in the Fourier domain and transforming both sides of(6.21) we have:| ˆφ(ω)| 2 = ˆψ(ω). (6.22)

✐✐154 Chapter 6. High-order SL approximation schemesNow, since ψ is a real and even function of y, its Fourier transform ˆψ is also a realand even function of ω. Moreover, the assumption ψ ∈ W 2,1 (R) implies that ˆψ(ω)is bounded and decays like O(ω −2 ) for ω → ±∞.Therefore, ˆψ being also nonnegative, its square root ˆψ(ω) 1/2 is real, even and nonnegative.In addition, ˆψ 1/2 is also bounded and decays like O(ω −1 ), and this impliesthat ˆψ 1/2 ∈ L 2 (R). Finally, looking at the inverse Fourier transform F −1 as an operatormapping L 2 (R) into L 2 (R), we obtain that the solution φ defined byφ(y) = F −1 { ˆψ(ω)1/2 } (6.23)is a well-defined even real function of L 2 (R) solving (6.21).In conclusion, the SL scheme (6.10) is L 2 stable, being equivalent to a stablescheme. Note that stability of LG schemes is intended in terms of the L 2 norm‖vh n‖ 2, but in our case this coincides with stability in the discrete norm ‖V n ‖ 2 . Infact, we have∫‖vh‖ n 2 2 = vh(x) n 2 dx =R∫ ( ) ⎛ ⎞∑= vi n φ i (x) ⎝ ∑ vj n φ j (x) ⎠ dx =R ij∫vi n vjn φ i (x)φ j (x)dx == ∑ i,j= ∑ iR(v n i ) 2 = 1∆x ‖V n ‖ 2 2where we have applied the orthogonality relationship (6.18). Thus, continuous anddiscrete norm coincide up to the scaling factor 1/∆x.Remark 6.2. Due to a theorem of Bochner (see [Sch99]), the functions havingpositive Fourier transform could also be characterized as positive semi-definitefunctions, defined as follows.Definition 6.3. A complex-valued function g : R d → R is said to be positivesemi-definite ifn∑ n∑a k g(x k − x j )ā j ≥ 0 (6.24)k=1 j=1for any x k ∈ R d , a k ∈ C (k = 1, . . . , n) and for all n ∈ N.Remark 6.4. We explicitly note that (6.23) needs not have (and in fact, has not ingeneral) a unique solution. Equating ˆψ and | ˆφ| 2 , we neglect any information aboutthe phase diagram of φ, and this in turn makes it possible to have multiple solutions.We will get back to this point when treating the case with variable coefficients.An example: P 1 interpolation Although the case of P 1 interpolation needsnot this kind of stability analysis, we start by treating this simple case, for which

✐✐6.1. Semi-Lagrangian schemes for the advection equation 155explicit computations can be performed. This will allow to illustrate some remarkablepoint, in particular the lack of uniqueness for the solution. We recall that inthis case the reference function has the form⎧⎪⎨ 1 + y if − 1 ≤ y ≤ 0ψ [1] (y) = 1 − y if 0 ≤ y ≤ 1(6.25)⎪⎩0 elsewherewhose Fourier transform of is:ˆψ [1] (ω) = 2 − 2 cos ωω 2 =( )sinω 22( ω) 2(6.26)2Now, taking the square root of this transform as in (6.23), we get∣∣sinˆφ ω ∣[1] 2(ω) =∣ ω ∣and accordingly,2(6.27){∣ } ∣sin ω ∣φ [1] (y) = F −1 2∣ ω ∣. (6.28)This solution, numerically computed by FFT, is shown along with the referencefunction (6.25) in the upper row of Figure 6.1.On the other hand, a different (and possibly, more natural) solution can be pickedup by noting that ˆψ [1] (ω) is also the squared magnitude of2ˆφ [1] (ω) = sin ω 2ω2whose inverse Fourier transform is explicitly computable as{φ [1] 1 if − 1/2 ≤ y ≤ 1/2(y) =0 elsewhere.(6.29)(6.30)This example shows more precisely the effect of losing uniqueness by neglecting thephase information – in fact, we can obtain an infinity of solutions to (6.21) whichdiffer in the phase term. This is not a major problem here, since at this stage weonly need existence of a solution. When dealing with variable coefficient equations,however, we will need a more careful choice of the solution.Lagrange interpolation of odd order As shown in Chapter 3, the generalform of the reference basis function for Lagrange interpolation of odd degree is⎧[r/2]+1∏ y − kif 0 ≤ y ≤ 1−kk≠0,k=−[r/2]⎪⎨ψ [r] (y) =.(6.31)r∏ y − kif [r/2] ≤ y ≤ [r/2] + 1−k⎪⎩ k=10 if y > [r/2] + 1

✐✐156 Chapter 6. High-order SL approximation schemesand extended by symmetry for y < 0. Since (6.31) is piecewise polynomial andcompactly supported, its second derivative is the sum of a bounded compactlysupported term, plus a finite number of Dirac masses, and hence ψ ∈ W 2,1 (R). Thesymbolic computation of the Fourier transforms (see [Fe10], [Fe11]) shows that, forall odd orders r ≤ 13, they have the structureˆψ [r] (ω) = a 0 + a 2 ω 2 + · · · + a r−1 ω r−1 (sin ω ) r+1,(6.32)ω r+1 2where the polynomial contains only positive terms of even degree (a 2m > 0, a 2m+1 =0). All Fourier transforms of the form (6.32) are therefore nonnegative. A naturalconjecture would be that this structure holds for any odd value of r, however ratherthan proving such a property, we simply note here that it holds for any order ofpractical interest.The basis functions ψ [r] (y) and the solutions φ [r] (y) = F −1 { ˆψ [r] (ω) 1/2 } for r = 3and r = 5 are shown in respectively the middle and the lower row of figure 6.1. Inthis case, direct and inverse transforms have been computed numerically by FFT.Finite element reconstructions In this situation, since finite element reconstructionsfail to be translation invariant (this point has already been remarked in Chapter3) the technique used for Lagrange reconstructions cannot be applied and we willrather perform a Von Neumann analysis. Unfortunately, due to the lack of translationinvariance, B is no longer a circulating matrix and therefore such analysis onlygives a necessary condition.Assume that (6.2) is set on the interval [0, 1] with periodic conditions, thatthe interval is split into N subintervals, the interpolation degree being r on eachsubinterval. The total number of nodes is n n = Nr and ∆x = 1/(Nr) is the spacestep between nodes. We assume that the Courant number λ = c∆t/∆x is in (0, 1),whereas the case of large Courant numbers can be recovered as a byproduct of thisanalysis, as it has been shown for the CIR scheme.In the j–th subinterval, the reconstruction depends on the values of the numericalsolution at the nodes x (j−1)r , . . . , x jr . Accordingly, the matrix B has theblock circulating structure⎛⎞B 1 0 0 · · · 0 0 B 2B 2 B 1 0 · · · 0 0 0B =0 B 2 B 1 · · · 0 0 0⎜· · · · · · · · · · · · · · · · · · · · ·.⎟⎝ 0 0 0 · · · B 2 B 1 0 ⎠0 0 0 · · · 0 B 2 B 1with B 1 , B 2 ∈ R r×r , and B 2 of the formB 2 = ( 0 | b ) ,b being a column vector of R r . More in detail, setting k = jr (that is, denoting byx k the right extremum of the j–th subinterval), we can write the j–th block of thescheme as⎧u ⎪⎨n+1k= l 0 (λ)u n k + · · · + l r(λ)u n k−r.(6.33)⎪⎩u n+1k−r+1 = l 0(λ + r − 1)u n k + · · · + l r(λ + r − 1)u n k−r

✐✐6.1. Semi-Lagrangian schemes for the advection equation 157linearcubicquinticFigure 6.1. Reference basis function for interpolation and equivalent LGreference basis functions obtained via (6.23), for linear (upper), cubic (middle) andquintic (lower) reconstructionwhere l 0 (·), . . . , l r (·) are the Lagrange basis functions expressed in a reference intervalwith unity step between nodes, and, by periodicity, u n 0 ≡ u n n nif j = 1. Notethat, formally, the sign of the reference variable in the l i is reversed with respectto the physical variable because of the positive sign of λ, which corresponds to anegative shift along characteristics.

✐✐158 Chapter 6. High-order SL approximation schemesNow, assuming that ρ is an eigenvalue for the eigenvector v, we have from (6.33):⎧[ l0 (λ) − ρ ] v k + · · · + l r (λ)v k−r = 0⎪⎨.(6.34)l 0 (λ + r − 1)v k + · · · +⎪⎩+ [ l r−1 (λ + r − 1) − ρ ] v k−r+1 + l r (λ + r − 1)v k−r = 0.Since (6.34) has r equations and r + 1 unknowns, it is possible, for example viaGaussian elimination, to express v k−r as a function of v k in the formv k−r = h(ρ, λ) v k (6.35)with h(·, ·) an iterated composition of rational functions. Then, repeating the procedurefor N subintervals and imposing the periodicity condition we obtain:h(ρ, λ) N = 1, (6.36)that is, h(ρ, λ) must be a N–th root of the unity. As it has been shown in theclassical case of circulating matrices, we can pass to the modulus, obtaining theequation of the curve containing the eigenvalues in the implicit form∣ h(ρ, λ)∣ ∣ = 1. (6.37)Note that this latter condition suits the case of N → ∞, in which the N-th rootsof the unity become dense on the unit circle of C. A different interpretation, whentreating an unbounded grid, is that (6.37) states that an eigenvector v should bebounded in l ∞ .Last, the Von Neumann condition is satisfied if:∣ h(ρ, λ)∣ ∣ = 1 implies that |ρ| ≤ 1. (6.38)We show in Fig. 6.2 the eigenvalues of the P 2 and P 3 schemes for differentvalues of λ, with respectively N = 25 (50 nodes) and N = 17 (51 nodes).The P 2 case We treat separately the P 2 case, for which the basis functionsin (6.33) have the form:l 0 (y) = 1 (y − 1)(y − 2)2l 1 (y) = y(2 − y) (6.39)l 2 (y) = 1 y(y − 1)2and allow for an explicit computation. More precisely, we have the followingTheorem 6.5. Consider the problem (6.2). Assume that the scheme S is in theblock form (6.33), with r = 2 and l 0 , l 1 , l 2 given by (6.39), that the space step ∆xis constant, and that 0 < λ < 1. Then, the scheme S satisfies condition (6.38).Proof. We refer to [P06] for the detailed computations, while sketching here onlythe main steps. Carrying out the computations outlined above, we obtain for (6.35)the expression:⎡⎤v k−2 = ⎣ ρ − l 0(λ) + l1(λ)l0(λ+1)ρ+l 1(λ+1)⎦ vl 2 (λ) − l1(λ)l2(λ+1) kρ+l 1(λ+1)

✐✐6.1. Semi-Lagrangian schemes for the advection equation 1590.80.60.40.20!0.2!0.4!0.6!0.8!1 !0.5 0 0.5 10.80.60.40.20!0.2!0.4!0.6!0.8!1 !0.5 0 0.5 1Figure 6.2. Eigenvalues of the P 2 (upper) and P 3 (lower) scheme forλ = 0.1, 0.2, . . . , 1.which, via some algebra, gives the condition:[ 2ρ 2 ] N+ (λ + 4)(λ − 1)ρ + (λ − 2)(λ − 1)= 1.λ(λ − 1)ρ + λ(λ + 1)Passing to the modulus, and writing ρ = z + iw ∈ C, we obtain the equationw 4 + [ 2z 2 + (λ 2 + 3λ − 4)z + (2λ 3 − λ 2 − 3λ + 2) ] w 2 ++ [ z 4 + (λ 2 + 3λ − 4)z 3 + (2λ 3 + λ 2 − 9λ + 6)z 2 ++(−5λ 2 + 9λ − 4)z − 2λ 3 + 3λ 2 − 3λ + 1 ] = 0.

✐✐160 Chapter 6. High-order SL approximation schemesBy working in the auxiliary variable w 2 we obtain, for z ∈ [1 − 2λ, 1], w 2 as thefollowing function of z and λ:withw 2 = −[ 2z 2 + (λ + 4)(λ − 1)z + (2λ 2 + λ − 2)(λ − 1) ] + √ D(z, λ)2D(z, λ) = λ 2 (λ − 1)(λ + 7)z 2 + 2λ 2 (λ − 1)(2λ 2 + 7λ − 7)z ++λ 2 (4λ 4 − 4λ 3 − 11λ 2 + 22λ − 7).Finally, for z ∈ [1 − 2λ, 1], it is easy to verify that w 2 < 1 − z 2 , which amounts toprove that eigenvalues are in the unit disc of C.Nonuniform space grids In this situation, if H j = r∆x j is the measure ofthe j–th subinterval, we can define local Courant numbersλ j = c∆t∆x j.We set again k = jr with j = 1, . . . , N, so that the j–th block of the scheme readsnow ⎧⎪ u n+1⎨ k= l 0 (λ j )u n k + · · · + l r(λ j )u n k−r.(6.40)⎪ ⎩u n+1k−r+1 = l 0(λ j + r − 1)u n k + · · · + l r(λ j + r − 1)u n k−r .Theorem 6.6. Assume that, for any j, 0 < λ min ≤ λ j ≤ λ max < 1. Assumemoreover that the constant step scheme (6.33) satisfies condition (6.38) for anyλ ∈ [λ min , λ max ]. Then condition (6.38) is also satisfied by the variable step scheme(6.40).Proof. Following the same procedure as above, condition (6.36) is now replaced byN∏h(ρ, λ j ) = 1. (6.41)j=1Keeping the analogy with the uniform mesh case, the condition which characterizesthe curve containing the eigenvalues may be rewritten asN∏|h(ρ, λ j )| = 1. (6.42)j=1Now, all the term in the product satisfy the boundsminj|h(ρ, λ j )| ≤ |h(ρ, λ j )| ≤ max |h(ρ, λ j )| (6.43)jso thatmin |h(ρ, λ j )| N ≤jN∏j=1|h(ρ, λ j )| ≤ max |h(ρ, λ j )| N (6.44)j

✐✐6.1. Semi-Lagrangian schemes for the advection equation 1610.80.60.40.20!0.2!0.4!0.6!0.8!1 !0.5 0 0.5 1Figure 6.3. Eigenvalues of the P 2 scheme for 0.014 ≤ λ j ≤ 0.77 and 50 nodes.and we can conclude that if condition (6.38) is satisfied with a uniform grid for anyλ ∈ [λ min , λ max ], then it is also satisfied in the nonuniform case.We show in Fig. 6.3 (in crosses) the eigenvalues of the P 2 scheme, obtainedwith 25 subintervals of random measure and 0.014 ≤ λ j ≤ 0.77, compared with (indots) the eigenvalues associated to the extreme values of the local Courant numbers.Remark 6.7. The stability analysis for the case of a nonuniform grid also coversthe case of variable coefficient equations, although in the situation of low Courantnumbers. The Von Neumann condition |ρ| ≤ 1 should be regarded in general asa necessary condition, since we have no informations on the normality of the matrixB. However, there is a strong evidence that this matrix is quasi-normal, thismeaning that in its Jordan decomposition (4.40), the condition number K(T ) =‖T −1 ‖ · ‖T ‖, although not unity, still remains uniformly bounded with respect to ∆.The general case for translation invariant reconstructionsWhen trying to extend to variable coefficient equations the stability analysis basedon the equivalence with LG schemes, it is clear that the relationship (6.21) canjustify replacing Galerkin projection with interpolation, but cannot account for thedeformation of the basis functions. On the other hand, if in (6.9) the approximateadvection X ∆ (ξ, t n+1 ; t n ) is replaced with a rigid translation ξ−x i +X ∆ (x i , t n+1 ; t n )along the approximate characteristic passing through x i , we obtain an approximate

✐✐162 Chapter 6. High-order SL approximation schemesLG scheme of the form∑∫v n+1iv n i∫φ i(ξ − xi + X ∆ (x i , t n+1 ; t n ) ) φ j (ξ)dξ.φ i (ξ)φ j (ξ)dξ = ∑iR i R(6.45)With some abuse of notation, we will refer to a scheme in the form (6.45) as anArea-weighted Lagrange–Galerkin (ALG) scheme. Schemes of this kind are used tocircumvent the problem of computing the integrals of deformed basis functions inexact LG schemes, and their stability analysis is relatively easy if the basis functionsφ k are piecewise polynomial and compactly supported, as usual in the finite elementsetting. Here, on the contrary, the φ k are obtained through a reference function φsolving (6.21), which is characterized in a somewhat weaker way by means of itsFourier transform.Thus, while the equivalence between (6.45) and (6.3) can be proved by thesame ideas of the constant coefficient case, an ad hoc proof is necessary to showthat (6.45) represents a “small” perturbation of an exact LG scheme, that is∥ B SL (∆; t n ) ∥ 2= ∥ B ALG (∆; t n ) ∥ 2≤ ∥ B LG (∆; t n ) ∥ 2+ C∆t. (6.46)In turn, this will be proved in the form∥ B ALG − B LG∥ ∥2≤ (∥ ∥ B ALG − B LG∥ ∥1 · ∥∥ B ALG − B LG∥ ∥∞) 1/2≤ C∆t, (6.47)which is satisfied if both the 1-norm and the ∞-norm of B ALG − B LG are boundedby an O(∆t).In order to suit the main situation of interest, i.e., Lagrange interpolation ofodd degree, the LG reference basis function will be assumed to be piecewise smooth,with at most a countable set of isolated discontinuities y k . More precisely, we willassume the following:• The function φ(y) satisfies the decay condition• Its derivative is in the form|φ(y)| ≤φ ′ (y) = φ ′ s(y) +C φ1 + |y| 3 (6.48)∞∑k=−∞w k δ(y − y k ), (6.49)where the regular and the singular part satisfy the further boundsand the singularities y k have the expression|φ ′ s(y)| ≤ C s1 + y 2 , (6.50)|w k | ≤ C w1 + k 2 , (6.51)y k = αk + β. (6.52)

✐✐6.1. Semi-Lagrangian schemes for the advection equation 163• The vectorfield f(·, t) is uniformly Lipschitz continuous, that isfor any t ∈ R.|f(x 1 , t) − f(x 2 , t)| ≤ L f |x 1 − x 2 | (6.53)• The approximation X ∆ (·) is consistent, that isz i = X ∆ (x i , t n+1 ; t n ) = x i − ∆t f(x i , t n+1 ) + O(∆t 2 ). (6.54)We give a couple of preliminary lemmas, which will be useful in order toestimate the perturbation appearing in (6.47).Lemma 6.8. For any a ∈ R + , b ∈ R,∫dyπ(1 + a)(1 + y 2 )(1 + (ay + b) 2 =) (1 + a) 2 + b 2 ; (6.55)R∞∑k=−∞for some positive constant C.1Cπ(1 + a)(1 + k 2 )(1 + (ak + b) 2 ≤) (1 + a) 2 + b 2 . (6.56)Proof. The integral (6.55) can be computed explicitly (the result has been obtainedby symbolic integration with Mathematica, see [Fe11]). The inequality (6.56) canbe obtained by observing that all terms of the series are bounded, and therefore itssum can be given an upper bound, up to a mutiplicative constant, by comparisonwith the corresponding integral (6.55).Lemma 6.9. Let assumptions (6.53), (6.54) hold. Then, for any c ∈ R + , d ∈ R,both series∞∑ 1c + (j − zi∆x + (6.57)d)2j=−∞∞∑i=−∞1c + (j − zi∆x + (6.58)d)2are uniformly bounded as functions of respectively the index i and the index j.Proof. Assume first that the smallest value attained by the term (j − zi∆x + d)2 isexactly zero.Consider the series (6.57). Let i (and therefore z i ) be fixed, and denote by j 0the index for which (j 0 − zi∆x + d)2 = 0 (that is, j 0 = zi∆x− d). Accordingly, we have∞∑j=−∞1c + (j − zi∆x + = d)2=∞∑j=−∞∞∑j=−∞1c + (j − j 0 ) 2 =1c + j 2

✐✐164 Chapter 6. High-order SL approximation schemeswhere the last form is obtained by an obvious shift in the summation index, and isindependent of the index i.Consider now (6.58), and let j be fixed. Denote by i 0 the index for which(j − zi 0∆x + d)2 = 0 (that is, zi 0∆x= j + d). By the consistency assumption (6.54), wehavez i = x i − ∆tf(x i ) + O(∆t 2 ) (6.59)z i0 = x i0 − ∆tf(x i0 ) + O(∆t 2 ) (6.60)so that using the Lipschitz continuity of f and the triangular inequality (in the formof a difference), we obtainand, for ∆t small enough,|z i − z i0 | = ∣ ∣x i − x i0 − ∆t(f(x i ) − f(x i0 )) + O(∆t 2 ) ∣ ∣ ≥i=−∞≥ |x i − x i0 | − ∆t|f(x i ) − f(x i0 )|(1 + O(∆t)) ≥≥ (1 − ∆tL f )|x i − x i0 |(1 + O(∆t)) (6.61)|z i − z i0 | ≥ 1 2 |x i − x i0 |. (6.62)Turning back to (6.58), we have, using (6.62), and again a shift in the summationindex:∞∑ 1c + (j − zi∆x + =∑∞ 1d)2 c + ( zi 0∆x − ≤ zi∆x )2≤==i=−∞∞∑i=−∞∞∑i=−∞∞∑i=−∞1c + 1 4 ( xi 0∆x − = xi∆x )21c + 1 4 (i 0 − i) 2 =1c + 1 4 i2 (6.63)which is independent of j.Lastly, if the smallest value for the term (j − zi∆x + d)2 is nonzero, then thesame arguments may be applied, unless for a fractional perturbation of the constantd. We note here that the sum of both series is continuous with respect to such aperturbation, so that it is possible to pass to the supremum, obtaining a uniformbound in this case as well. The technical details are left to the reader.We present now the main result. It states the stability of the Area-weightedLG scheme under assumptions which generalize the parallel result in [MPS88]. Thiswill allow to apply the theory to LG basis functions obtained by solving (6.21).Theorem 6.10. Let the basic assumptions (6.17) and (6.48)–(6.54) hold, andassume x j = j∆x. Then, there exists a positive constant C independent of n, ∆xand ∆t such that, at a generic step n ∈ N and for ∆t small enough,∥ B ALG (∆; t n ) − B LG (∆; t n ) ∥ 2≤ C∆t.Proof. The proof uses the bound (6.47), and is split in several steps.

✐✐6.1. Semi-Lagrangian schemes for the advection equation 165Step 1. We first need to estimate the difference |X ∆ (ξ)−(ξ−x i +X ∆ (x i , t n+1 ; t n ))|.Denoting for shortness X ∆ (ξ, t n+1 ; t n ) as X ∆ (ξ) and again X ∆ (x i , t n+1 ; t n ) as z i ,by (6.53) and (6.54) we obtain:|X ∆ (ξ) − (ξ − x i + z i )| ≤ C X |ξ − x i |∆t. (6.64)At a later time, we will use again the change of variables η = ξ/∆x − i. In this newvariable, (6.64) is rewritten asStep 2.∣ bALGij|X ∆ ((η + i)∆x) − (η∆x + z i )| ≤ C X |η|∆x∆t. (6.65)We proceed to estimate |b ALGij (∆; t n )−b LGij (∆; t n)|. By definition, we have∣∫∫∣∣∣ (∣ij = φ j (ξ − x i + z i ))φ i (ξ)dξ − φ j X ∆ (ξ) ) φ i (ξ)dξ∣ ≤− b LGR≤ 1∆x∫R( ξ −∣ φ xi + z i∆xR)− j − φ( )∣ X ∆ ( )∣(ξ) ∣∣∣∆x − j ·ξ ∣∣∣∣ φ ∆x − i dξ.Now, introducing the variable η = ξ/∆x − i, we obtain∫( ) ( ∣ bALGij − b LG ∣ij ≤ η∆x +∣ φ ziX ∆ ((η + i)∆x) ∣∣∣− j − φ− j)∣· |φ(η)| dη ≤R ∆x∆x∫∫ X ∆ ((η+i)∆x)∆x −j≤φ ′ (y)dyη∆x+zR ∣i· |φ(η)| dη ≤∆x −j∣∫≤ |φ ′ (y)| · |φ(η)| dydηDwhere D is defined byand y(η), y(η) byD = { (η, y) : η ∈ R, y ∈ [ y(η), y(η) ]}( η∆x + ziy(η) = min∆x), X∆ ((η + i)∆x)− j∆x( η∆x + ziy(η) = max, X∆ ((η + i)∆x)∆x ∆x)− j.Note that the integration domain is modified so as to have a y-projection of positivemeasure for any η.Since the integrand is positive, we can give a further bound on this integral byenclosing the integration domain in a larger domain on which the integrand could bebounded in a more straightforward way. By (6.65), the domain D can be includedin a set of the formE ={(η, y) : η ∈ R, y ∈[η − C X ∆t|η| + z i∆x − j, η + C X∆t|η| + z i∆x − j ]}Figure 6.4 shows the inclusion of the integration domain D in the set E definedabove. In the figure, the liney = η∆x + z i∆x− j

✐✐166 Chapter 6. High-order SL approximation schemesyDE1C X t z i x − j1−C X t z i x − jz i x − jFigure 6.4. Inclusion of the integration domain in the set E.and the curvey = X∆ ((η + i)∆x)∆x− jof the η−y plane, enclose the dark-shaded domain D, which in turn appears includedin the light-shaded set E.We have therefore:∣ bALGij∫− b LG ∣ ≤ij∫≤∫=D|φ ′ (y)| · |φ(η)| dydη ≤∫ η+CX ∆t|η|+ z i∆x −jR η−C X ∆t|η|+ z i∫ η+CX ∆t|η|+ z i∆x −jR∫+η−C X ∆t|η|+ z i∫ η+CX ∆t|η|+ z i∆x −jRη−C X ∆t|η|+ z i∆x −j |φ ′ (y)| · |φ(η)| dydη =∆x −j |φ ′ s(y)| · |φ(η)| dydη +k=−∞∆x −j [ ∞∑|w k |δ(y − y k )]· |φ(η)| dydη= A ij + B ij . (6.66)

✐✐6.1. Semi-Lagrangian schemes for the advection equation 167Step 3. Estimation of A ij . Using the decay informations on φ and φ ′ s, we canestimate A ij as:∫A ij =R≤ C s C φ∫∫ η+CX ∆t|η|+ z i∆x −jη−C X ∆t|η|+ z i∆x∫ −jη+CX ∆t|η|+ z i∆x −j 1R η−C X ∆t|η|+ z i∆x −j∫≤ 2C s C φ C X ∆tR|φ ′ s(y)|dy |φ(η)| dη ≤1 + y 2 dy 11 + |η| 3 dη ≤11 + ( (1 − C X ∆t)η + zi∆x − j) 2|η|dη, (6.67)1 + |η|3where the inner integral has been estimated by multiplying the sup of the functionto be integrated by the measure of the interval. Now, it is easy to see that we canbound the second term of the integrand in (6.67) as|η|1 + |η| 3 ≤ 21 + η 2 ,so that, applying lemma 6.8 and collecting all multiplicative constants in a singleconstant C A , we obtain∫1A ij ≤ 4C s C φ C X ∆tR 1 + ( 1(1 − C X ∆t)η + zi∆x − j) 21 + η 2 dη =π(2 − C X ∆t)= 4C s C φ C X ∆t(2 − C X ∆t) 2 + ( z i∆x − j) 2 ≤≤C A ∆t1 + ( z i∆x − j) 2(6.68)where the last inequality has been obtained assuming also that ∆t is small enough.Step 4.Estimation of B ij . The integral B ij may be splitted as∫ ∫ [η+CX ∆t|η|+ z i∆x −j ∞]∑B ij =|w k |δ(y − y k ) · |φ(η)| dydη =R η−C X ∆t|η|+ z i∆x −j k=−∞∞∑∫ ∫ η+CX ∆t|η|+ z i∆x −j= |w k |δ(y − y k ) · |φ(η)| dydη. (6.69)R η−C X ∆t|η|+ z i∆x −jk=−∞Note that the measure to be integrated has its support on the segment of intersectionbetween the line y = y k and the set E. Denoting by η − kthe abscissa of theintersection of the line y = y k with the boundary y = (1 + C X ∆t)η + zi∆x − j (thisrepresents the extremum of smallest magnitude for this segment), and vice versaη + kfor the boundary y = (1 − C X∆t)η + zi∆x− j (extremum of largest magnitude),we have (see Figure 6.5)|η + k − η− k | = 2C X∆t1 − C 2 X ∆t2 ∣ ∣∣yk− z i∆x + j ∣ ∣∣ == 2C X∆t1 − C 2 X ∆t2 ∣ ∣∣αk + β −z i∆x + j ∣ ∣∣

✐✐168 Chapter 6. High-order SL approximation schemesFigure 6.5. Integration of the singular part of φ ′ on the set E.and|η − k | = 1∣∣y k − z ∣i ∣∣1 + C X ∆t ∆x + j ==11 + C X ∆tso that for ∆t small enough we also have∣∣αk + β − z i∆x + j ∣ ∣∣|η + k − η− k | ≤ 4C ∣X∆t ∣αk + β − z ∣i ∣∣∆x + j ,|η − k | ≥ 1 2∣∣αk + β − z i∆x + j ∣ ∣∣ .Now, using such bounds and again the decay estimate for φ as in step 3, we obtain∫R∫ η+CX ∆t|η|+ z i∆x −jη−C X ∆t|η|+ z i∆x −j∣ αk + β −z i∆xδ(y − y k ) · |φ(η)| dydη ≤ 2C X ∆t+ j∣ ∣1 + ∣ αk + β −z i∆x + j∣ ∣ 3 ≤≤ 4C X ∆t11 + ( αk + β − zi∆x + j) 2

✐✐6.1. Semi-Lagrangian schemes for the advection equation 169and therefore∞∑1B ij ≤ 4C X ∆t |w k |k=−∞ 1 + ( αk + β − zi∆x + j) 2 ≤∞∑ C w1≤ 4C X ∆t1 + k 2 k=−∞ 1 + ( αk + β − zi∆x + j) 2 ≤C B ∆t≤(1 + α) 2 + ( β − zi∆x + j) 2(6.70)where we have applied againg Lemma 6.8 and collected all multiplicative constantinto C B .Conclusions. To sum up, recalling (6.66) and applying Lemma 6.9 to the estimates(6.68) and (6.70), we obtain that∥∥B ALG (∆; t n ) − B LG (∆; t n ) ∥ ∑∞≤ sup (A ij + B ij ) ≤ C∆t,i∥ B ALG (∆; t n ) − B LG (∆; t n ) ∥ ∑1≤ sup (A ij + B ij ) ≤ C∆t,jso that the bound (6.47) holds.ijRemark 6.11. It is immediate to see that, according to this result, the class ofbasis functions for which the Area-weighted LG scheme is stable includes all bounded,piecewise Lipschitz continuous functions with bounded support. In this respect, thisresult includes and generalizes the corresponding theorem in [MPS88].Application to Lagrange interpolation We turn back to the case of Lagrangeinterpolation of odd order. As we said, the solution of (6.21) is not explicitly known,but rather characterized in terms of its Fourier transform. So what is needed is torelate properties (6.48)–(6.52) to properties of the Fourier transform ˆφ(ω).• Concerning the rate of decay of φ(y), it is known that φ decays like |y| −kprovided ˆφ (k) ∈ L 1 (R). Therefore, assumption (6.48) is satisfied if ˆφ ′′′ ∈L 1 (R).• Checking assumption (6.49) requires to single out a periodic component iniω ˆφ(ω) (that is, in the transform F[φ ′ ]). This component generates the singularpart (sum of evenly spaced Dirac distributions) in the inverse Fouriertransform. Once defined iω ˆφ s as the difference between iω ˆφ and its periodiccomponent, the decay assumption (6.50) requires its second derivative withrespect to ω to be in L 1 (R).• Assumption (6.51) is related to the decay of Fourier coefficients for periodicfunctions. It turns out that this assumption is satisfied provided the periodiccomponent of iω ˆφ(ω) has locally L 1 second derivative with respect to ω.

✐✐170 Chapter 6. High-order SL approximation schemesExample: P 1 interpolation We re-read this example, already solved in theconstant coefficient case, by discussing the use of the two different solutions of theintegral equation (6.21). First, using (6.23), we get∣ sinω ∣ˆφ [1] 2(ω) =∣ ω ∣(6.71)and the solution could be defined accordingly as the inverse transform of (6.71) (seethe upper line of figure 6.1). Although this solution would be fine for the constantcoefficient case, it does not satisfy the required decay assumptions. For example, thetransform (6.71) is not three times differentiable, even in the sense of distributions,so it is not expected to satisfy (6.48).On the other hand, using the other possibility, that is,and taking the inverse transform as2ˆφ [1] (ω) = sin ω 2ω2{ sinω}φ [1] (y) = F −1 2ω=2, (6.72){1 if − 1/2 ≤ y ≤ 1/20 elsewhere.we obtain a solution which satisfies all the basic assumptions, as it is easy to see.Note that, even in lack of an explicit inverse transform, this could be extractedfrom the properties of the Fourier transform (6.72). First, (6.72) defines an analyticfunction, so the inverse transform φ [1] (y) decays faster than any negative power ofy and (6.48) is satisfied (actually, it has bounded support). Second, the transformof φ [1]′ readsF{φ [1]′} = 2i sin ω 2 .Here, we only have a periodic component and φ [1]′s ≡ 0, so (6.50) is satisfied. In turn,the periodic component is also analytic, so its Fourier coefficients have a fast decayand (6.51) holds (in fact, the only nonzero coefficient is related to the “harmonic”sin(ω/2), and φ [1]′ (y) = δ(y + 1/2) − δ(y − 1/2)).Lagrange interpolation of higher (odd) order Although, for r ≥ 3, theproblem of differentiability of ˆφ [r] outlined for r = 1 does not appear any longer,defining a solution through (6.23) would not allow to single out a periodic componentin F[φ ′ ], and as a result the solution in the y-domain would have a slower decay. Away to circumvent this problem is to define a solution in the ω-domain as√ˆφ [r] a0 + a 2 ω(ω) =2 + · · · + a r−1 ω r−1· sin ω ∣∣sin ω r−12∣ =2 2ω|ω| r−12= A r (ω) · B r (ω). (6.73)In (6.73), A r and B r denote respectively the algebraic and the trigonometric termof ˆφ [r] . Note that in (6.73) all derivatives are continuous at ω = 0, whereas due tothe structure of B r (ω) it occurs that derivatives are continuous only up to a certainorder (depending on r) for ω coinciding with multiples of 2π.

✐✐6.1. Semi-Lagrangian schemes for the advection equation 171FTo check the basic assumptions, we also need to compute F{√φ [r]′} (ω) = iω ˆφ [r] a0 + a 2 ω(ω) = i2 + · · · + a r−1 ω r−1· sin ω|ω| r−122{φ [r]′} :∣∣sin ω 2(6.74)which contains again an algebraic and a periodic term. The first one has a finiteimaginary limit i √ a r−1 for ω → ±∞, so that we can rewrite (6.74) as the sum of aterm vanishing for ω → ±∞, plus a periodic term giving the asymptotic behaviour,that isF{φ [r]′} (ω) = i(√ )a0 + a 2 ω 2 + · · · + a r−1 ω r−1− √ a r−1 sin ω ∣∣sin ω ∣2 2|ω| r−12+i √ a r−1 sin ω ∣∣sin ω ∣2 2= iC r (ω) + iD r (ω).r−12As it has been explained at the start of the section, in order to satisfy the basicassumptions we have to check that(i) The transform (6.73) has its third derivative (w.r.t. ω) in L 1 (R);(ii) The term C r (ω) has its second derivative in L 1 (R);(iii) The term D r (ω) has a locally L 1 second derivative.Let us start with point (i). We have, from elementary differentiation rules:=d 3dω 3 ˆφ [r] (ω) = A ′′′r B r + 3A ′′r B ′ r + 3A ′ rB ′′r + A r B ′′′r . (6.75)Now, the terms B r , B r ′ and B r ′′ are always bounded. The term B r′′′ is bounded forall r > 3, whereas, for r = 3, it has the form of a sequence of Dirac distributions of= O(1/ω 2 ), so thatthe right-hand side of (6.75) is in L 1 (R) and point (i) is satisfied for any r ≥ 3.Concerning point (ii), using the same arguments of point (i), it suffices that thealgebraic term in brackets, along with its first and second derivative, be O(1/ω 2 ).This is in fact the case, so the requirement of point 2 is also satisfied. Lastly, pointconstant weight. On the other hand, we have that A r , . . . , A ′′′r(iii) is also trivially satisfied since D r ′′ is bounded on R. All symbolic computationsrequired for this subsection, along with the Fourier transforms, have been carriedout with Mathematica for r ≤ 13 (a sample of the related code is listed in [Fe11]).We show in Figure 6.6 the reference LG basis functions obtained by solvingnumerically (6.21) by Fast Fourier Transform, both in the form (6.23) (left column)and in the form (6.73) (right column), for the linear, cubic and quintic Lagrangeinterpolation. Note that, since D r has a period of 4π in the ω-domain, the discontinuitiesin φ [r] (that is, Dirac distributions in φ [r]′ ) appear in the y-domain atmultiples of 1/2, so that (6.52) holds with α = 1/2, β = 0 (in fact, even harmonicsare missing in D r , so w k = 0 for k even). If r−12is even (e.g., in the linear andquintic case), thensin ω ∣∣sin ω r−1 (2∣ = sin ω ) r+12,2 22so that D r (ω) contains a finite number of harmonics, and as a result the LG referencebasis function has a finite number of discontinuities.∣r−12r−12,+

✐✐172 Chapter 6. High-order SL approximation schemeslinearcubicquinticFigure 6.6. Equivalent LG reference basis functions obtained via (6.23)(left) and (6.73) (right), for linear (upper), cubic (middle) and quintic (lower) reconstruction6.1.4 ConvergenceWe sum up the convergence analysis in a theorem. Note that the technical assumptions(6.53)–(6.54) are already included as assumptions on smoothness of f andconsistency rate of time discretization.Theorem 6.12. Let f, g ∈ W p,∞ , u be the solution of (6.1) and v n j be definedby (6.3), with an interpolation basis ψ i satisfying the assumptions of Theorem 6.1.

✐✐6.1. Semi-Lagrangian schemes for the advection equation 173Assume moreover that (5.55), (5.56), (6.4) hold, and that the function φ, solutionof (6.21), satisfies (6.48)–(6.52). Then, for any j ∈ Z and n ∈ [1, T/∆t],∣ vnj − u(x j , t n ) ∣ ∣ → 0 (6.76)as ∆t → 0, ∆x = o ( ∆t 1/r) .Moreover, if u ∈ L ∞ ([0, T ], W s,∞ (R)), then)‖V n − U(t n )‖ 2≤ C(∆t p + ∆xmin(s,r+1). (6.77)∆tWe also discuss here modified equation and dispersive behaviour of the SLscheme, along with the interactions between discretization steps at the level oftuning of the scheme.Numerical dispersionRepeating the analysis of chapter 5, we consider again the equation in the form (6.2),a scheme in the form (6.10), and assume v to be a regular extension of the numericalsolution of the scheme. If we replace the error estimate for P 1 interpolation with thecomplete estimate for Lagrange interpolation of a generic order r, we can expressthe value I r [V n ](x j − c∆t) asI r [V n ](x j − c∆t) = v(x j − c∆t, t n ) −∏(x j − c∆t − x k )x k ∈S(r + 1)!∂ r+1∂x r+1 v(ξ j),where S is the stencil of points involved in the reconstruction, and ξ j is an unknownpoint located between the extreme points of of S. Next, retracing the computationsof the first-order case, we obtain from(6.10):[ ]∂r+1v(x j , t n+1 ) = v(x j − c∆t, t n ) + ν(∆)∂x r+1 v(x j − c∆t) + O(∆x)which is in the form (4.85), with k = r + 1, once set the dispersivity coefficient ν asν(∆) = −∏(x j − c∆t − x k )x k ∈S(r + 1)! ∆t(note that sgn(ν(∆)) = (−1) r−12 , and that all terms of the product are O(∆x)). Weobtain therefore a modified equation in the form (4.84), and more preciselyv t + cv x = (−1) r−12 cr∆x r+1∆t∂ r+1 ( ) ∆xr+1∂x r+1 v + o , (6.78)∆twith c r a bounded nonnegative function of λ ′ . Then, the dispersivity of the schemecan be analysed by applying standard self-similarity arguments to the modifiedequation. This procedure shows that the resolution of discontinuities is of orderO(∆x/∆t 1/(r+1) ).

✐✐174 Chapter 6. High-order SL approximation schemesInterplay between discretization stepsAs it has been noticed treating the CIR scheme, the convergence estimate (6.77)shows an interaction between discretization steps, which requires to relate ∆x and∆t in a less trivial form with respect to usual difference schemes. Once expressed∆t as a function of ∆x in the form ∆t = ∆x α , we briefly discuss the optimizationof the convergence rate in (6.77), as well as the resolution of discontinuities, withrespect to the parameter α.Maximization of the convergence ratesteps in (6.77), we obtainUsing the relationship between the two(‖V n − U(t n )‖ 2≤ C ∆x αp + ∆x min(s,r+1)−α) .Since the first exponent increases with α, whereas the second decreases, the convergencerate is maximized when they coincide, that is, forα =min(s, r + 1). (6.79)p + 1Note that the ∆t/∆x relationship depends on the regularity.smooth (more precisely, if s > r + 1), then we obtainIf the solution isα = r + 1p + 1 ,a choice which clearly maximizes the consistency rate. Note that, in principle, ifr > p (e.g. coupling a second-order time integration with a cubic reconstruction),the relationship optimizing the consistency rate entails a vanishing Courant number.As the regularity decreases, α decreases, which means that less regular solutions arebetter approximated with larger time steps.Resolution of singularities In analysing the behaviour of the scheme on discontinuoussolutions, still two opposite effects appear: on one hand, large Courantnumbers cause a lesser number of reconstructions to be performed (this causing asmaller numerical dispersion), on the other hand, they cause larger errors in computingcharacteristics (this causing a less precise location of singularities).Assuming as before that ∆t = ∆x α , the analysis of the modified equationshows that the dispersion of discontinuities is of the order of ∆x 1−α/(r+1) . Hence,taking into account that characteristics are computed with an error of ∆x αp , it turnsout that the resolution of the scheme is maximized when the two orders coincide,that is, with the choicer + 1α =p(r + 1) + 1 . (6.80)Note that this relationship implies α < 1/p, so that ∆x = o(∆t p ), and in practiceα ≈ 1/p (for example, for a second-order time and space discretization p = r = 2,and we would obtain α = 3/7). In general, this choice could lead to relatively largetime discretization errors. However, it confirms the indication that high Courantnumbers are more suitable to treat nonsmooth solutions.

✐✐6.1. Semi-Lagrangian schemes for the advection equation 1756.1.5 ExamplesIn order to compare the performances of the various schemes, we consider again thesimple one-dimensional problem (5.66), which is rewritten here:{u t (x, t) − (x − ¯x)u x (x, t) = 0 (x, t) ∈ (0, 1) × (0, 1)u(x, 0) = u 0 (x).(6.81)As in Chapter 5, we have set ¯x = 1.1 and used three initial conditions u 0 of differentregularity with bounded support. The first is the characteristic function (5.67), thesecond is a W 1,∞ function in the form:u 0 (x) = max(sin(2πx), 0), (6.82)and the third is in W 2,∞ :u 0 (x) = max(sin(2πx), 0) 2 . (6.83)Note that, in addition to what has been remarked in Chapter 5, the initial conditionshave been changed to have a non-polynomial structure, so as to evaluate theefficiency of the reconstruction procedure.The test has been carried out combining a second-order Runge–Kutta schemeto follow characteristics with P 2 finite element, cubic and 3rd-order ENO reconstructions.∆t/∆x relationships in the form ∆t = ∆x α have been considered forthree different values of α: the first, α = 1/2, is close to the value optimizing theresolution of singularities, the second is the classical linear relationship α = 1 (moreprecisely, ∆t = 5∆x following the choice made for the monotone case) and the thirdis the value of α optimizing the consistency rate (in the P 2 case, this criterion givesagain α = 1). The ENO scheme has been tested on the most efficient relationshipused in the cubic case.We list in Table 6.1 numerical errors and overall convergence rates in the 2-norm for the various schemes. Figure 6.7 compares the schemes on the advectionof the characteristic function (5.67). Here, the first three figures are obtained withα = 1/2 which (nearly) optimizes the resolution of discontinuities, whereas thefourth is obtained with α = 4/3 which optimizes the consistency rate for the cubicreconstruction.As a first general remark, note that the error tables confirm the general indicationthat less regular solutions are better approximated with large Courantnumbers, i.e., with lower values of α. In cubic approximation, the relationshipmaximizing the consistency rate proves to be relatively inefficient, partly because itwould require a smoother solution (in W 4,∞ ) to achieve its best performance. Thisis also apparent from Figure 6.7, which shows the remarkable increase in numericaldispersion caused by the choice α = 4/3. Note also that the difference in convergencerates should be considered against complexity. Under the relationships shownin the tables, a test with 100 nodes requires 10 time steps with α = 1/2, 20 withα = 1 and 464 with α = 4/3. In terms of absolute error, the best performancesare obtained by the P 2 scheme – but we point out once more that the cubic schemewould require a smoother solution to increase its convergence rate.

✐✐176 Chapter 6. High-order SL approximation schemesL ∞ solutionP 2 cubic ENO3n n α = 1/2 α = 1 α = 1/2 α = 1 α = 4/3 α = 1/225 1.44 · 10 −1 1.44 · 10 −1 1.43 · 10 −1 1.43 · 10 −1 2.35 · 10 −1 2.1 · 10 −150 6.82 · 10 −2 5.96 · 10 −2 1.01 · 10 −1 1.04 · 10 −1 1.29 · 10 −1 1.06 · 10 −1100 5.38 · 10 −2 5.52 · 10 −2 6.6 · 10 −2 7.74 · 10 −2 9.39 · 10 −2 8.22 · 10 −2200 4.09 · 10 −2 4.09 · 10 −2 5.36 · 10 −2 6.06 · 10 −2 7.65 · 10 −2 5.94 · 10 −2400 2.57 · 10 −2 3.56 · 10 −2 3.72 · 10 −2 4.63 · 10 −2 5.83 · 10 −2 4.39 · 10 −2rate 0.62 0.50 0.49 0.41 0.50 0.56W 1,∞ solutionP 2 cubic ENO3n n α = 1/2 α = 1 α = 1/2 α = 1 α = 4/3 α = 1/225 2.04 · 10 −2 2.04 · 10 −2 8.37 · 10 −2 8.37 · 10 −2 1.30 · 10 −1 8.94 · 10 −250 9.92 · 10 −3 9.62 · 10 −3 2.03 · 10 −2 2.44 · 10 −2 4.99 · 10 −2 2.58 · 10 −2100 3.44 · 10 −3 3.15 · 10 −3 7.21 · 10 −3 9.11 · 10 −3 1.83 · 10 −2 1.01 · 10 −2200 1.82 · 10 −3 1.98 · 10 −3 3.26 · 10 −3 4.39 · 10 −3 8.18 · 10 −3 4.47 · 10 −3400 7.96 · 10 −4 7.92 · 10 −4 1.29 · 10 −3 1.87 · 10 −3 3.59 · 10 −3 1.86 · 10 −3rate 1.17 1.17 1.50 1.37 1.29 1.40W 2,∞ solutionP 2 cubic ENO3n n α = 1/2 α = 1 α = 1/2 α = 1 α = 4/3 α = 125 1.80 · 10 −2 1.80 · 10 −2 1.06 · 10 −1 1.06 · 10 −1 1.40 · 10 −1 1.06 · 10 −150 5.76 · 10 −3 3.70 · 10 −3 1.96 · 10 −2 2.01 · 10 −2 6.47 · 10 −2 2.56 · 10 −2100 2.72 · 10 −3 1.19 · 10 −3 3.84 · 10 −3 4.32 · 10 −3 1.65 · 10 −2 5.99 · 10 −3200 1.35 · 10 −3 2.96 · 10 −4 1.41 · 10 −3 9.83 · 10 −4 3.38 · 10 −3 1.64 · 10 −3400 6.63 · 10 −4 8.42 · 10 −5 6.67 · 10 −4 2.49 · 10 −4 7.65 · 10 −4 4.73 · 10 −4rate 1.19 1.93 1.83 2.18 1.88 1.95Table 6.1. Errors in the 2-norm, P 2 , cubic and 3rd-order ENO schemes,second-order time discretization.6.2 Semi-Lagrangian schemes for the convex HJequationIn order to illustrate the convergence theory of high-order Semi-Lagrangian schemesfor Hamilton–Jacobi equations, we go back to our one-dimensional model problem:{u t (x, t) + H(u x (x, t)) = 0 (x, t) ∈ R × [0, T ]u(x, 0) = u 0 (x) x ∈ R.(6.84)Throughout this section, we will make the standing assumption that H is a strictlyconvex function, and more precisely:H ′′ (p) ≥ m H . (6.85)

✐✐6.2. Semi-Lagrangian schemes for the convex HJ equation 177Figure 6.7. Numerical results for the advection of a characteristic functionobtained via P 2 (upper left), cubic (upper right), 3rd-order ENO (lower left)and cubic with maximum consistency rate (lower right) schemes, second-order timediscretization, 200 nodes.We recall that this implies that, if u 0 is a Lipschitz continuous function, then u(x, t)is semiconcave for all t > 0, and if u 0 is semiconcave itself, then u(x, t) is uniformlysemiconcave for t ≥ 0. Moreover, in terms of the Legendre transform, condition(6.85) implies thatH ∗′′ (α) ≤ 1m H. (6.86)6.2.1 Construction of the schemeAs it has been shown for the linear case, turning to high-order the first-order SLscheme (5.92) only requires to replace the monotone P 1 interpolation I 1 with areconstruction of degree r > 1:{vn+1j= minα∈R [∆tH∗ (α) + I r [V n ](x j − α∆t)]v 0 j = u 0(x j ).(6.87)In this section, we will develop a convergence theory which allows to treat the casesof Lagrange, finite elements, ENO and WENO reconstructions. Strictly speaking,the scheme (6.87) is not monotone, but in fact Barles–Souganidis theorem requiresa weaker condition, that is, that (6.87) should be monotone up to a term o(∆t).As we will show, this generalized form of monotonicity can occur under increasing

✐✐178 Chapter 6. High-order SL approximation schemesCourant numbers if the scheme is Lipschitz stable, so this will be the main stabilityissue.6.2.2 ConsistencyConsistency analysis clearly follows the same lines of the low-order scheme, once theP 1 reconstruction is replaced by a high-order version, and the definition of S SL ischanged accordingly. We can repeat the steps of the corresponding estimate, unlessfor using (6.4) in place of (5.43), obtaining at last for the high-order SL scheme(6.87) the consistency estimate∥∥L SL (∆; t, U(t)) ∥ ≤ C ∆xr+1. (6.88)∆tRemark 6.13. As it has already been noticed about the first-order SL scheme,due to the fact that characteristics are straight lines, the bound (6.88) does nottake into account the accuracy of time discretization. This is related to the specificstructure of the HJ equation (6.84), but if a transport term appears in the equation,and is discretized with order p, then the consistency estimate coincides with the oneobtained in the linear case, that is,∥ L SL (∆; t, U(t)) ∥ ()≤ C ∆t p + ∆xr+1.∆t6.2.3 Lipschitz stabilityIn order to be as general as possible, we start with some basic assumption on thereconstruction operator. First, we define the stencil of reconstruction asS(x) = (x − h − ∆x, x + h + ∆x), (6.89)so that the reconstruction at the point x uses the nodes inside S(x). For example,a quadratic Lagrange reconstruction can be performed taking one node on the leftand two nodes on the right of the point x (in this case, h − = 1, h + = 2), or twonodes on the left and one on the right (and in this case, h − = 2, h + = 1). In asecond-order finite element or ENO/WENO reconstruction, both cases are possible,thus h − = h + = 2. A third-order Lagrange reconstruction is performed using twonodes on the left and two on the right, so that h − = h + = 2. In the third-orderfinite element or ENO/WENO case, h − = h + = 3, and so forth.Then, we assume that, given a Lipschitz continuous function v(x) and thesequence V = {v j } j∈I = {v(x j )} j∈I , the operator I r [V ] satisfies:|I r [V ](x) − I 1 [V ](x)| ≤ C r maxx j−1,x j,x j+1∈S(x) |v j+1 − 2v j + v j−1 | (C r < 1). (6.90)The key point of assumption (6.90) consists in requiring that C r < 1. In fact,as it is immediate to check, any polynomial reconstruction satisfies (6.90) with aconstant depending on the interpolation degree r. Besides a technical reason whichwill become clear in the proof of Lemma 6.14, the bound on C r amounts to assumethat I r is not “too sensitive” with respect to large second increments which canoccur on singularities of the solution.

✐✐6.2. Semi-Lagrangian schemes for the convex HJ equation 179It will also be useful to define the functionF j (α) := ∆tH ∗ (α) + I r [V n ](x j − α∆t), (6.91)and denote again by ᾱ j the value of α achieving the minimum in (6.87) and in F j .In order to stress the “locality” in the above definition, the notation explicitly showsthe dependence of F on the node index j, although for simplicity the dependenceon the time step n has been dropped.We give now a bound on the second increment of the numerical solution. Thebound is globally one-sided, but becomes two-sided at the foot of characteristics,that is, in a neighborhood U(x j + ᾱ j ∆t), with U defined asU(x) = (x − h∆x, x + h∆x), (6.92)with a fixed h > max(h + , h − ), so that S(x) ⊂ U(x). More precisely, we have thefollowing technical result:Lemma 6.14. Let V n be defined by the scheme (6.87). If (6.86) holds, then, forany j ∈ Z and n ≥ 1,vj+1 n − 2vj n + vj−1 n ≤∆x2m H ∆t . (6.93)Moreover, assuming, in addition, that (6.90) holds, thenmax |vi+1 n − 2vi n + vi−1| n ≤ C ∆x2x i−1,x i,x i+1∈U(x j+ᾱ j∆t)∆t(6.94)with U defined by (6.92) and for some positive constant C depending on C r , h, andm H .Proof. We start by proving (6.93). By (6.87) we have, for n ≥ 1,v n j= ∆tH ∗ (ᾱ j ) + I r[Vn−1 ] (x j − ᾱ j ∆t),vj−1 n = ∆tH ∗ [(ᾱ j−1 ) + I r Vn−1 ] (x j−1 − ᾱ j−1 ∆t) ≤(≤ ∆tH ∗ ᾱ j − ∆x )[+ I r Vn−1 ] (x j − ᾱ j ∆t),∆tvj+1 n = ∆tH ∗ [(ᾱ j+1 ) + I r Vn−1 ] (x j+1 − ᾱ j+1 ∆t) ≤(≤ ∆tH ∗ ᾱ j + ∆x )[+ I r Vn−1 ] (x j − ᾱ j ∆t)∆t(where the inequalities come from the use of ᾱ j ± ∆x/∆t instead of the actualminimizers ᾱ j±1 ), so that(vj+1 n − 2vj n + vj−1 n ≤ ∆t[H ∗ ᾱ j + ∆x ) (− 2H ∗ (ᾱ j ) + H ∗ ᾱ j − ∆x )]≤∆t∆t( ) 2 ∆x≤ ∆t sup H ∗′′ , (6.95)∆twhich, using (6.86), gives (6.93).

✐✐180 Chapter 6. High-order SL approximation schemesIn order to prove (6.94), we take into consideration the values of F j (α) forx j − α∆t coinciding with a grid node, that is for α = k∆x/∆t. Let k j be definedas:⌊k j = sgn (ᾱ j ) |ᾱ j | ∆t ⌋∆xso that{x j − ᾱ j ∆t ∈ [x j−kj , x j−kj+1) if ᾱ j ≤ 0x j − ᾱ j ∆t ∈ (x j−kj−1, x j−kj ] if ᾱ j ≥ 0Since the two cases are perfectly symmetric, we examine in detail the second, ᾱ j ≥ 0.In this case, the point x j − ᾱ j ∆t is on the left of x j , and more preciselyMoreover, the minimizer ᾱ j satisfiesx j−kj−1 < x j − ᾱ j ∆t ≤ x j−kj ≤ x j .k j∆x∆t ≤ ᾱ j < (k j + 1) ∆x∆t .We will denote by δ j the (signed) maximal second increment of the numerical solutionin the neighborhood U(x j − ᾱ j ∆t) and by j − k j + l j the index of the node atwhich this second increment occurs. If δ j ≥ 0, then (6.94) follows from (6.93) andwe have nothing else to prove. We will assume, therefore, that δ j < 0, so thatδ j = −maxx i−1,x i,x i+1∈U(x j−ᾱ j∆t)|v n i+1 − 2v n i + v n i−1| == v n j−k j+l j+1 − 2v n j−k j+l j+ v n j−k j+l j−1. (6.96)We recall that |l j | < h, and will also assume in what follows that l j ≥ 0, the reversecase being similar to prove.Step 1.Upper bound of the second increment of the function F j . We have:(F j (k + 1) ∆x∆t= ∆t≤) (− 2F j k ∆x ) (+ F j (k − 1) ∆x )=∆t∆t( [H ∗ (k + 1) ∆x ) (− 2H ∗ k ∆x∆t∆t+ v n j−k−1 − 2v n j−k + v n j−k+1 ≤)+ H ∗ ((k − 1) ∆x∆t)]+∆x2m H ∆t + vn j−k−1 − 2vj−k n + vj−k+1 n ≤ 2∆x2m H ∆t , (6.97)where the second increment of H ∗ has been estimated as in (6.95), and the lastinequality holds for n ≥ 1.Step 2.In the second step, we prove that( ) (∆xF j (ᾱ j ) ≥ min[F j k j , F j (k j + 1) ∆x )]− ∆x2∆t∆t 8m H ∆t − C r|δ j |. (6.98)

✐✐6.2. Semi-Lagrangian schemes for the convex HJ equation 181In fact, set ᾱ j = (1 − ¯θ)k j ∆x/∆t + ¯θ(k j + 1)∆x/∆t, with ¯θ ∈ [0, 1]. Taking intoaccount the uniform convexity of H ∗ implied by (6.86), we have( ) (H ∗ (ᾱ j ) ≥ (1 − ¯θ)H ∗ ∆xk j +∆t¯θH ∗ (k j + 1) ∆x )−∆t− 1 ( ) 22 ¯θ(1 − ¯θ) ∆xsup H ∗′′ ≥∆t(≥ (1 − ¯θ)H ∗ ∆xk j∆t) (+ ¯θH ∗ (k j + 1) ∆x )−∆t−∆x28m H ∆t 2 (6.99)in which the third term has been minimized with respect to ¯θ.In a similar way, we also get, using (6.90) and the inclusion of S(x) in U(x),I r [V n ](x j − ᾱ j ∆t) ≥ I 1 [V n ](x j − ᾱ j ∆t) −−C r maxx |vn i+1 − 2vi n + vi−1| n ≥i−1,x i,x i+1∈S(x j−ᾱ j∆t)≥ I 1 [V n ](x j − ᾱ j ∆t) −−C rmaxx i−1,x i,x i+1∈U(x j−ᾱ j∆t)= (1 − ¯θ)v n j−k j+ ¯θv n j−k j−1 − C r |δ j | =(= (1 − ¯θ)I r [V n ]+¯θI r [V n ]x j − k j∆x∆t ∆t )+|v n i+1 − 2v n i + v n i−1| =(x j − (k j + 1) ∆x∆t ∆t )− C r |δ j |,which, combined with (6.99), gives( ) (F j (ᾱ j ) ≥ (1 − ¯θ)F ∆xj k j +∆t¯θF j (k j + 1) ∆x )− ∆x2∆t 8m H ∆t − C r|δ j |,which in turn implies (6.98).Step 3. Upper bound of the increment of F j between k j ∆x/∆t and (k j +1)∆x/∆t.Assume first that F j (k j ∆x/∆t) < F j ((k j + 1)∆x/∆t). Using (6.98) and the optimalityof ᾱ j , we obtain(F j (k j − 1) ∆x ) ( )∆x≥ F j (ᾱ j ) ≥ F j k j − ∆x2∆t∆t 8m H ∆t − C r|δ j |,that is, considering the extreme terms(∆xF j k j∆t)− F j((k j − 1) ∆x∆t)≤ ∆x28m H ∆t + C r|δ j |.On the other hand, from (6.97) written with k = k j , we also have(F j (k j + 1) ∆x ) ( ) ( ) (∆x∆x− F j k j ≤ F j k j − F j (k j − 1) ∆x )+ 2∆x2∆t∆t∆t∆t m H ∆t

✐✐182 Chapter 6. High-order SL approximation schemesand therefore, combining the last two inequalities, we get the desired bound on thefirst increment:(F j (k j + 1) ∆x ) ( )∆x− F j k j ≤ 17∆x2∆t∆t 8m H ∆t + C r|δ j |. (6.100)Moreover, if F j ((k j + 1)∆x/∆t) < F j (k j ∆x/∆t), then(F j (k j + 1) ∆x ) ( )∆x− F j k j ≤ 0∆t∆tand (6.100) is also trivially satisfied.Step 4. We show that, in order for ᾱ j to be a minimizer for F j , δ j must satisfy(6.94). To this end, we first assume that F j (k j ∆x/∆t) ≤ F j ((k j + 1)∆x/∆t) andbound the values F j (k∆x/∆t) from above using a function ˜F j (k∆x/∆t) constructedso as to coincide with F j at k j ∆x/∆t and to have first and second increments greateror equal to the corresponding increments of F j . Taking into account the boundson the first and second increment of F j obtained in the previous steps of the proof,we could define ˜F j so that its first increment between k j ∆x/∆t and (k j + 1)∆x/∆twould be given by the right-hand side of (6.100), and the second increment wouldbe obtained in view of (6.96), (6.97) as(˜F j (k + 1) ∆x ) (− 2∆t˜F j k ∆x ) (+∆t˜F j (k − 1) ∆x )=∆t⎧2∆x 2⎪⎨if k ≠ k j + l j ,m H ∆t=∆x ⎪⎩2m H ∆t + δ j if k = k j + l j .(6.101)We recall that, as it is easy to prove by induction, if a sequence f i has aconstant second increment,f i+2 − 2f i+1 + f i ≡ d,then the values of the elements and of the first increments are given byf k+l = f k + l(f k+1 − f k ) + (1 + 2 + · · · + (l − 1))d,f k+l+1 − f k+l = (f k+1 − f k ) + ld.Using the previous equalities, a function ˜F j suitable for our purpose could be

✐✐6.2. Semi-Lagrangian schemes for the convex HJ equation 183defined more explicitly as(∆x˜F j k j∆t) (∆x= F j k j∆t),(˜F j (k j + 1) ∆x ) ( )∆x= F j k j + 17∆x2∆t∆t 8m H ∆t + C r|δ j |,˜F j((k j + 2) ∆x∆t)= F j(k j∆x∆t)+ 2( ) 17∆x28m H ∆t + C r|δ j | + 2∆x2m H ∆t ,(.˜F j (k j + l j ) ∆x ) ( ) ( )∆x 17∆x2= F j k j + l j∆t∆t 8m H ∆t + C r|δ j |+(1 + · · · + (l j − 1)) 2∆x2m H ∆t ,(˜F j (k j + l j + 1) ∆x ) ( ) ( )∆x 17∆x2= F j k j + l j∆t∆t 8m H ∆t + C r|δ j | ++ (1 + · · · + (l j − 1)) 2∆x2m H ∆t +( ) 17∆x2+8m H ∆t + C 2∆x 2r|δ j | + l jm H ∆t + δ j(note that the computation of ˜Fj ((k j + l j + 1)∆x/∆t) is “restarted” because of(6.101)) and last, for any integer m > 0,(˜F j (k j + l j + m) ∆x ) ( )∆x= F j k j +∆t∆t( )17∆x2+ (l j + m)8m H ∆t + C r|δ j | ++ (1 + · · · + (l j + m − 1)) 2∆x2m H ∆t + mδ j ≤( ) ( )∆x17∆x2≤ F j k j + (h + m)∆t8m H ∆t + C r|δ j | ++ (1 + · · · + (h + m − 1)) 2∆x2m H ∆t + mδ j. (6.102)On the other hand, by (6.98) and the optimality of ᾱ j , we also have, for any m > 0,(˜F j (k j + l j + m) ∆x ) (≥ F j (k j + l j + m) ∆x )≥∆t∆t≥ F j (ᾱ j ) ≥( ) (∆x≥ min[F j k j , F j (k j + 1) ∆x )]−∆t∆t−∆x28m H ∆t − C r|δ j |. (6.103)We explicitly note that in (6.103) the first inequality follows from the construction+

✐✐184 Chapter 6. High-order SL approximation schemesof ˜F j as an upper bound for F j , the second one from the optimality of ᾱ j in F j ,and the third one from the lower bound (6.98).Recalling that we have assumed F j (k j ∆x/∆t) ≤ F j ((k j + 1)∆x/∆t) and δ j 0 of the left-hand side(which is in fact a maximum) is O(∆x 2 /∆t), provided C r < 1. Using the reverseestimate (6.93), we get at last (6.94).If otherwise F j ((k j +1)∆x/∆t) < F j (k j ∆x/∆t), then it is possible to redefinethe function ˜F j so that ˜F j ((k j + 1)∆x/∆t) = F j ((k j + 1)∆x/∆t), and the sameupper bounds are used on the first and second increments. In this way, we replace(6.102) with(˜F j (k j + l j + m) ∆x ) (= F j (k j + 1) ∆x∆t∆t)+( )17∆x2+ (l j + m − 1)8m H ∆t + C r|δ j | ++ (1 + · · · + (l j + m − 1)) 2∆x2m H ∆t + mδ j,which again implies (6.104). This construction completely parallels the previousone and is therefore left to the reader.Before proving Lipschitz continuity, we derive a couple of useful consequencesof Lemma 6.14. First, given a Lipschitz continuous function v(x) and the sequenceV = {v j } j∈I = {v(x j )} j∈I , the condition|I r [V ](x) − I 1 [V ](x)| = O(∆x),typical in the interpolation of a function with bounded derivative, is written by(6.90) as|I r [V ](x) − I 1 [V ](x)| ≤ C r maxx i−1,x i,x i+1∈S(x) |v i+1 − 2v i + v i−1 | ≤≤ C r (sup |v i+1 − v i | + sup |v i−1 − v i |) ≤≤ 2L∆x. (6.105)

✐✐6.2. Semi-Lagrangian schemes for the convex HJ equation 185Also, at the foot of a characteristic the stronger condition|I r [V n ](x i + ᾱ j ∆t) − I 1 [V n ](x i + ᾱ j ∆t)| ≤ C r C ∆x2∆t(6.106)holds for any j ∈ Z, n ≥ 1 and for any node i = j ± 1. In fact, (6.106) follows from(6.90) and Lemma 6.14, since U(x j + ᾱ j ∆t) contains all the nodes involved in thereconstructions I r [V n ](x j±1 + ᾱ j ∆t).We can now state the result of Lipschitz stability.Theorem 6.15. Let V n be defined by the scheme (6.87). Assume that (6.86),(6.90) hold, that ∆x = O(∆t 2 ), and that u 0 is Lipschitz continuous with Lipschitzconstant L 0 . Then, V n satisfy, for any i and j, the discrete Lipschitz estimate|v n i − vn j ||x i − x j | ≤ Lfor a constant L independent of ∆x and ∆t, and for 0 ≤ n ≤ T/∆t, as ∆t → 0.Proof. It clearly suffices to prove the claim for i, j such that i = j ± 1. Assumethat at the previous step the discrete solution satisfies|v n−1i− v n−1j |∆x≤ L n−1 .Making the argmin explicit and using (6.106), we havev n j= ∆tH ∗ (ᾱ j ) + I r[Vn−1 ] (x j + ᾱ j ∆t) ≥≥ ∆tH ∗ (ᾱ j ) + I 1[Vn−1 ] (x j + ᾱ j ∆t) − C r C ∆x2∆t . (6.107)In order to estimate the discrete incremental ratio of V n , we give on v n iv n i= ∆tH ∗ (ᾱ i ) + I r[Vn−1 ] (x i + ᾱ i ∆t) ≤≤ ∆tH ∗ (ᾱ j ) + I r[Vn−1 ] (x i + ᾱ j ∆t) ≤the bound≤ ∆tH ∗ (ᾱ j ) + I 1[Vn−1 ] (x i + ᾱ j ∆t) + C r C ∆x2∆t , (6.108)which results from both the optimality of ᾱ i and (6.106), and holds for any n ≥ 2.If n = 1, applying (6.105) instead of (6.106), we obtainv 1 i = ∆tH ∗ (ᾱ i ) + I r[V0 ] (x i + ᾱ i ∆t) ≤≤ ∆tH ∗ (ᾱ j ) + I r[V0 ] (x i + ᾱ j ∆t) ≤≤ ∆tH ∗ (ᾱ j ) + I 1[V0 ] (x i + ᾱ j ∆t) + 2L 0 ∆x. (6.109)From (6.107) and (6.108) we obtain, for n ≥ 2, the unilateral estimatev n i − vn j∆x≤ 1∆x(I1[Vn−1 ] (x i + ᾱ j ∆t) − I 1[Vn−1 ] (x j + ᾱ j ∆t) ) + 2C r C ∆x2∆t≤≤ L n−1 + 2C r C ∆x∆t , (6.110)

✐✐186 Chapter 6. High-order SL approximation schemesin which we have used the fact that the first-order reconstruction I 1 at step n − 1has also Lipschitz constant L n−1 . Interchanging the role of ᾱ j and ᾱ j+1 , we get thereverse estimatev n j − vn i∆x≤ L n−1 + 2C r C ∆x∆t ,and therefore|vj n − vn i | ≤ L n−1 + 2C r C ∆x∆x∆t . (6.111)A similar computation yields, for n = 1,|vj 1 − v1 i | ≤ L 0 + 4L 0 = 5L 0 , (6.112)∆xso that combining (6.111) with (6.112) and iterating back, we haveL n ≤ L n−1 + 2C r C ∆x∆t ≤ · · · ≤≤ L 1 + T − ∆t 2C r C ∆x∆t ∆t ≤≤ 5L 0 + 2T C r C ∆x∆t 2 . (6.113)Last, it is possible to get a finite limit in (6.113), if and only if ∆x =O(∆t 2 ).Applications to various reconstruction operatorsIn order to check the applicability of this theory to reconstruction operators of practicalinterest, we first test assumption (6.86) on the situation of an interpolatingpolynomial with stencil including the reconstruction point x (this is directly applicableto the case of finite element and ENO interpolations). Then, starting withthis result, we discuss the cases of symmetric Lagrange and WENO interpolations.Finite element and ENO interpolations Here, we make no assumptions on thestructure of the stencil S(x), unless that it must include one node on each side ofx. The reconstruction is assumed to be in the Newton formI r [V ](x) = V [x j0 ] + V [x j0 , x j1 ](x − x j0 ) + · · · ++ V [x j0 , . . . , x jr ](x − x j0 ) · · · (x − x jr−1 ), (6.114)where x j0 , . . . , x jr are r+1 adjacent nodes so that max(x j0 , . . . , x jr )−min(x j0 , . . . , x jr )= r∆x and, moreover,x ∈ (min(x j0 , . . . , x jr ), max(x j0 , . . . , x jr )) ⊂ S(x). (6.115)This definition includes both finite element interpolations, for which the reconstructionstencil is fixed (but not necessarily symmetric around x), and ENO reconstructions,for which it depends on the solution itself. The divided differences are defined,as usual, byV [x j0 ] = v j0 ,..V [x j0 , . . . , x jk ] = V [x j 1, . . . , x jk ] − V [x j0 , . . . , x jk−1 ]x jk − x j0(k = 1, . . . , r).

✐✐6.2. Semi-Lagrangian schemes for the convex HJ equation 187Note that, although in principle the nodes x j0 , . . . , x jk need neither to be adjacentnor to satisfy x ∈ (min x ji , max x ji ), it is possible to reorder the nodes so that bothconditions would be satisfied.According to its definition, we can bound a generic k-th order divided differenceas follows:|V [x j0 , . . . , x jk ]| ≤ 2 max |V [x j i, . . . , x jk+i−1 ]|k∆x(6.116)in which the max is performed for x ji , . . . , x jk+i−1 ∈ S(x). To prove (6.90), we startfrom the second divided differenceand, hence,|V [x j0 , x j1 , x j2 ]| = |v j 0− 2v j1 + v j2 |2∆x 2 ,|V [x j0 , x j1 , x j2 , x j3 ]| ≤ 2 max |v j i− 2v ji+1 + v ji+2 |3!∆x 3.|V [x j0 , . . . , x jr ]| ≤ 2r−2 max |v ji − 2v ji+1 + v ji+2 |r!∆x r .Plugging such bounds into (6.114), we get an estimate in the form (6.90); that is,where|I r [V ](x) − I 1 [V ](x)| ≤ |V [x j0 , x j1 , x j2 ](x − x j0 )(x − x j1 ) + · · · ++V [x j0 , . . . , x jr ](x − x j0 ) · · · (x − x jr−1 )| ≤≤ |v j 0− 2v j1 + v j2 |2∆x 2 M 2 ∆x 2 + · · · ++ 2r−2 max |v ji − 2v ji+1 + v ji+2 |r!∆x rM r ∆x r ≤r∑ M k 2 k−2≤ max |v ji − 2v ji+1 + v ji+2 |, (6.117)k!k=2M k := 1∆x k max |(x − x j 0) · · · (x − x jk−1 )| =x∈(x j0 ,x jk−1 )= max |t(t − 1) · · · (t − k + 1)|.t∈(0,k−1)It remains to check that C r < 1; that is,r∑ M k 2 k−2< 1. (6.118)k!k=2Indeed, the first values M k may be either computed by algebraic manipulations(up to M 5 ), or estimated by simply plotting the polynomials t(t − 1) · · · (t − k + 1)on the interval [0, k − 1]. It turns out that M 2 = 1/4, M 3 = 2 √ 3/9 ≈ 0.3849,M 4 = 1, M 5 ≈ 3.6314, M 6 < 17, and so on. Accordingly, the computation of theleft-hand side of (6.118) for various values of r gives C 2 = 1/8 = 0.125, C 3 ≈ 0.2533,C 4 ≈ 0.42, C 5 ≈ 0.6621, C 6 ≈ 1.04. We can conclude that by this technique, finiteelement and ENO reconstructions up to the fifth order can be proved to satisfy(6.90).

✐✐188 Chapter 6. High-order SL approximation schemesSymmetric Lagrange and WENO interpolations It is possible to extend thistheory to Lagrange and WENO interpolations, which work with a symmetric reconstructionstencil, that is, h + = h − . Assume that the interpolation I r [V ](x) isobtained as a sumI r [V ] = W 1 (x)P 1 (x) + · · · + W q (x)P q (x) (6.119)in which the P i are interpolating polynomials constructed on smaller stencils whichinclude the point x and, once x is fixed, the W i (x) are coefficients of a convexcombination (W i (x) ≥ 0, ∑ i W i(x) = 1). Then, we havemin(P 1 (x), . . . , P q (x)) ≤ I r [V ](x) ≤ max(P 1 (x), . . . , P q (x)).If deg P i ≤ 5, all the polynomials P i satisfy (6.90) and therefore, being satisfied byall P i , (6.90) is also satisfied by I r [V ].Note that (6.119) is the form for both symmetric Lagrange and WENO interpolation.In the first case, the functions W i play the role of linear weights asin (3.24), and it has been proved in Theorem 3.10 that they are positive and haveunity sum. In the second case, the form of I r is given by (3.27) and, although thenonlinear weights are no longer polynomials, they still perform a convex combinationdue to the positivity of the linear weights. Therefore, (6.90) is again satisfiedprovided deg P i ≤ 5. In both cases, this means a reconstruction of order r ≤ 9.6.2.4 ConvergenceLast, we present the main convergence result for the scheme (6.87).Theorem 6.16. Let u be the solution of (6.84), and V n be defined by the scheme(6.87). Assume that (6.105), (6.106) hold, that ∆x = O(∆t 2 ), and that u 0 isLipschitz continuous. Then,for 0 ≤ n ≤ T/∆t, as ∆t → 0.‖I r [V n ] − U(t n )‖ ∞ → 0Proof. First, note that the scheme is invariant for the addition of constants andconsistent. Then, retracing the proof of (5.103) and using (6.105) and Theorem 6.15,we obtain that the scheme is monotone up to a term O(∆x) (therefore, o(∆t)), andthe generalized monotonicity condition (4.58)–(4.59) is satisfied. Then, convergencefollows from Barles–Souganidis theorem.Remark 6.17. In this theorem, the condition ∆x = O(∆t 2 ) is only required toensure Lipschitz stability. In fact (and numerical experiments confirm this idea)this could be an overly restrictive assumption, possibly owing to some intrinsiclimitations in the technique of proof. Once ensured uniform Lipschitz continuityof numerical solutions, the generalized monotonicity condition requires instead theweaker constraint ∆x = o(∆t).

✐✐6.3. Commented references 189Before singularityn n cubic ... WENO3 WENO525 2.52 · 10 −3 1.29 · 10 −350 8.77 · 10 −5 1.87 · 10 −5100 1.53 · 10 −5 9.13 · 10 −7200 9.63 · 10 −7 2.01 · 10 −8rate 3.78 5.32After singularityn n cubic ... WENO3 WENO525 2.88 · 10 −3 3.05 · 10 −350 5.12 · 10 −5 5.83 · 10 −6100 2.19 · 10 −6 7.25 · 10 −8200 2.39 · 10 −7 1.89 · 10 −9rate 4.52 6.87Table 6.2. Errors before and after the singularity for Test (6.120),WENO3 and WENO5 schemes.6.2.5 ExamplesUniformly semiconcave solution This test (also considered, for example, in [LT00],[JP00]) concerns the HJ equation:{u t (x, t) + 1 2 (u x(x, t) + 1) 2 = 0(6.120)u(x, 0) = u 0 (x) = − cos(πx),on the interval [0, 2], with periodic boundary conditions. The solution eventuallydevelops a singularity in the derivative, and the approximate solution is computedbefore the singularity, at T = 0.8/π 2 , with 4 time steps, and after the singularity, atT = 1.5/π 2 , with 5 time steps. Plots of exact vs. approximate solutions are shownin Figure 6.8 and L ∞ errors obtained with different schemes are compared in Table6.2.Note that, according to the consistency estimate (6.88), the convergence ratesshould be r + 1 = 4 for the WENO3 scheme and r + 1 = 6 for the WENO5 scheme,since the time step has been kept constant. Roughly speaking, this behavior isconfirmed by numerical tests. Due to semiconcavity, even after the onset of the singularitythe computation of the approximate solution remains remarkably accurate,more than what could be deduced from consistency analysis for a nondifferentiablesolution. The influence of semiconcavity on the accuracy of numerical schemes hasalready been discussed in the presentation of numerical examples of Chapter 5.6.3 Commented referencesThe first attempts of a convergence analysis for high-order Semi-Lagrangian schemeshave appeared in a somewhat fragmentary form. As far as we can say, the first papercollecting convergence issues for (linear) SL schemes is [FF98]. Even this paper,

✐✐190 Chapter 6. High-order SL approximation schemes1’Initial_condition_test_1’10.8’Exact_solution’’3rd_order’0.60.50.40.200-0.2-0.4-0.5-0.6-0.8-1-10 0.5 1 1.5 20.80.6-1.20 0.5 1 1.5 2’Exact_solution’’3rd_order’0.40.20-0.2-0.4-0.6-0.8-1-1.20 0.5 1 1.5 2Figure 6.8. Approximate vs. exact solution for Test (6.120): initial condition(upper left), solution before singularity (upper right) and solution after singularity(lower).however, does not solve the point of l 2 stability in a completely rigorous way: aFourier analysis is performed, but Von Neumann condition is only enforced graphically,whereas an explicit (and very technical) solution of the Von Neumann analysisfor Lagrange reconstructions has been given later in [BM08].Equivalence between SL and Lagrange–Galerkin schemes has proved to be a simplertool for proving l 2 stability. This approach has been first studied in [Fe10]for constant coefficient equations, and extended to variable coefficient equations in[Fe11]. Basic references on the Lagrange–Galerkin technique are the seminal papers[DR82] and [Pi82], while the area-weighted version is proposed and studied in[MPS88]. The simplified analysis for finite element reconstructions has been developedin [P06] (the paper [FP07] gives a synthesis of this work), whereas an extensiveanalysis of the interaction between discretization steps is presented in [FFM01].High-order SL schemes for HJ equations have been first considered in [FF02],and a convergence analysis based on the condition ∆x = O(∆t 2 ) is carried out in[Fe03]. Convergence is directly proved in this paper starting from Lipschitz stability,whereas we have chosen here to prove it via the Barles–Souganidis Theorem. Theadaptation of the theory to weighted ENO reconstructions is given in [CFR05], alongwith a number of numerical tests comparing the various high-order versions of thescheme. Other numerical tests, mostly in higher dimension and concerned withapplications to front propagation and optimal control, are presented in [CFF04].

✐✐Chapter 7Control and GamesOne of the most typical applications of the theory of Hamilton–Jacobi equations isin the field of optimal control problems and differential games. Via the DynamicProgramming Principle (DPP in the sequel), formulated by R. Bellman in the 60s,many optimal control problems can be characterized by means of the associatedvalue function, which can be shown in turn to be the unique viscosity solution of apartial differential equation of convex Hamilton–Jacobi type, usually called Bellmanequation or Dynamic Programming equation.Although the direct numerical solution of the Bellman equation presents seriouscomplexity problems related to the dimension of the state space (the so-calledcurse of dimensionality), it also allows to recover a sharper information. Actually,in comparison with the more conventional technique given by the Pontryagin MaximumPrinciple (which characterizes the optimal control related to a given initialstate as a function of time), the Bellman equation provides a synthesis of feedbackcontrols, i.e. of controls expressed as a function of the state variable (instead oftime).In this chapter we present algorithms of Semi-Lagrangian type for the approximatesolution of Dynamic Programming equations related to optimal controlproblems and differential games. In fact, the same approach can be applied to theanalysis and approximation of zero-sum differential games where, under suitableconditions, the value function of the game solves a non-convex Hamilton–Jacobitype equation, usually called Isaacs equation.The Semi-Lagrangian numerical methods presented here are strongly connectedto Dynamic Programing for two main reasons. The first is that both Bellmanand Isaacs equations are derived themselves by passing to the limit in the DynamicProgramming Principle, the second is that the SL schemes are obtained by a discreteversion of the DPP, which plays the role of the Hopf–Lax formula for thisspecific case. This strategy of construction gives a nice control interpretation of theschemes and also helps in their theoretical analysis. Moreover, it allows to constructat the same time the approximations of both value function and optimal feedback,the latter being a practical solution to the original optimization problem.While the construction and theoretical analysis of SL schemes for DynamicProgramming equations allows for an elegant and general setting, their applicationto real-life industrial problems still suffers from an impractical computationalcomplexity. However, the continuous improvement of computers, as well as the191

✐✐192 Chapter 7. Control and Gamesdevelopment of faster and more accurate algorithms, make this field an active andpromising research area.7.1 Optimal control problems: first examplesIn order to present the main ideas, we start with the classical infinite horizonproblem. Let a controlled dynamical system be given by{ẏ(s) = f(y(s), s, α(s))(7.1)y(t 0 ) = x 0 ,where x 0 , y(s) ∈ R d , α : [t 0 , T ] → A, A ⊆ R m , T finite or +∞. In the sequel, wewill possibly use the less general formulation{ẏ(s) = f(y(s), α(s))(7.2)y(t 0 ) = x 0 .In the applications, it is unreasonable to assume a priori any regularity of the controlα. We will rather assume that the control is measurable, in which case the existenceand uniqueness for the solution of (7.1) is ensured by the Carathèodory theorem:Theorem 7.1. (Carathèodory) Assume that:(i) f(·, ·, ·) is continuous;(ii) there exists a positive constant L f > 0 such that|f(x, t, a) − f(y, t, a))| ≤ L f |x − y|,for all x, y ∈ R d , t ∈ R + and a ∈ A;(iii) f(x, t, α(t)) is measurable with respect to t.Then,is the unique solution of (7.1)∫ sy(s) = x 0 + f(y(τ), τ, α(τ))dτ (7.3)t 0Note that the solution is continuous, but only a.e. differentiable, so it mustbe regarded as a weak solution of (7.1). Note also that we are not interested hereto the most general version of this theorem, in which a condition of local Lipschitzcontinuity of f still allows to define a local solution y. This generalization is not ofgreat interest in Control Theory, in which a controlled dynamical system is usuallyrequired to have a solution for all s ≥ t 0 .By the theorem above, once fixed a control in the set of admissible controlsα ∈ A := {a : [t 0 , T ] → A, measurable} (7.4)there exists a unique trajectory y x0,t 0(s; α) of (7.1). Changing the control policy thetrajectory will change, and we will have a family of solutions of the controlled system(7.1) depending on α. To simplify notations, when considering the autonomousdynamics (7.2) and the initial time t 0 = 0, we will denote this family as y x0 (s; α)

✐✐7.1. Optimal control problems: first examples 193(or even with y(s) if no ambiguity occurs). Moreover, it is customary in DynamicProgramming to use the notations x and t instead of x 0 and t 0 (x and t will appearas variables in the Hamilton–Jacobi–Bellman equation), and therefore we willpermanently adopt this latter notation in the sequel.The second ingredient in the definition of an optimal control problem is thecost functional to be minimized. We introduce the problem by describing a coupleof classical examples.7.1.1 The infinite horizon problemOptimal control problems require the introduction of a cost functional J : A → Rwhich is used to select the “optimal trajectory” for (7.2). In the case of the infinitehorizon problem, we set t 0 = 0, x 0 = x, and this functional is defined asJ x (α) =∫ ∞0g(y x (s, α), α(s))e −λs ds (7.5)for a given λ > 0. The function g represents the running cost and λ is the discountfactor which allows to compare the costs at different times by rescaling the costs atthe initial time. From a technical point of view, the presence of the discount factorensures that the integral is finite whenever g is bounded.The goal of optimal control theory is to find an optimal pair (y ∗ , α ∗ ) whichminimizes the cost functional. If we look for optimal controls in open-loop form, i.e.as functions of t, then the main tool is given by the Pontryagin Maximum Principle,which gives the necessary conditions to be satisfied by an optimal couple (y ∗ , α ∗ ).A major drawback of an open-loop control is that, being constructed as a functionof time, it cannot take into account errors in the real state of the system, due forexample to model errors or external disturbances, which may bring the evolutionfar from the optimal forecasted trajectory. Another limitation of this approach isthat it requires to compute again the optimal control for any different initial state,so that it becomes impossible to design a static controller for the system.For these reasons, we are interested in the so-called feedback controls, thatis, controls expressed as functions of the state of the system. Under an optimalfeedback control, if the system trajectory is modified by an external perturbation,then the system reacts by changing its control strategy according to the change inthe state. One of the main motivations to use the Dynamic Programming approachis precisely that it allows to characterize the optimal feedback, as we will see laterin this chapter.The starting point of Dynamic Programming is to introduce an auxiliary function,the value function, which in the case of the infinite horizon problem is definedasv(x) = infα∈A J x(α), (7.6)where, as above, x is the initial position of the system. The value function has aclear meaning: it is the optimal cost associated to the initial position x. This isa reference value which can be useful to evaluate the efficiency of a control – forexample, if J x (ᾱ) is close to v(x), this means that ᾱ is “efficient”.Bellman’s Dynamic Programming Principle gives a first characterization ofthe value function.

✐✐194 Chapter 7. Control and GamesProposition 7.2 (DPP for the infinite horizon problem). Under the assumptionsof Theorem 7.1, for all x ∈ R d and τ > 0,{∫ τ}v(x) = inf g(y x (s; α), α(s))e −λs ds + e −λτ v(y x (τ; α)) . (7.7)α∈A0Proof. Denote by ¯v(x) the right-hand side of (7.7). First, we remark that, for anyx ∈ R d and ᾱ ∈ A,J x (ᾱ) ===≥∫ ∞0∫ τ0∫ τ0∫ τ0g(ȳ(s), ᾱ(s))e −λs ds =g(ȳ(s), ᾱ(s))e −λs +∫ ∞τg(ȳ(s), ᾱ(s))e −λs ds =∫ ∞g(ȳ(s), ᾱ(s))e −λs + e −λτ g(ȳ(s + τ), ᾱ(s + τ))e −λs ds ≥g(ȳ(s), ᾱ(s))e −λs + e −λτ v(ȳ(τ))(here, y x (s, ᾱ) is denoted for shortness as ȳ(s)).extreme terms of the inequality, we get0Passing to the infimum in thev(x) ≥ ¯v(x) (7.8)To prove the opposite inequality, we recall that ¯v is defined as an infimum, so that,for any x ∈ R d and ε > 0, there exists a control ᾱ ε (and the corresponding evolutionȳ ε ) such that¯v(x) + ε ≥∫ τ0g(ȳ ε (s), ᾱ ε (s))e −λs ds + e −λτ v(ȳ ε (τ)). (7.9)On the other hand, the value function v being also defined as an infimum, for anyx ∈ R d and ε > 0 there exists a control ˜α ε such thatusing (7.10) into (7.9), we get¯v(x) ≥∫ τ0v(ȳ ε (τ)) + ε ≥ Jȳε(τ)(˜α ε ) (7.10)g(ȳ ε (s), ᾱ ε (s))e −λs ds + e −λτ Jȳε(τ)(˜α ε ) − (1 + e −λτ )ε ≥≥ J x (̂α) − (1 + e −λτ )ε ≥≥ v(x) − (1 + e −λτ )ε, (7.11)where ̂α is a control defined bŷα(s) ={ᾱ ε (s) 0 ≤ s < τ˜α ε (s − τ) s ≥ τ.(7.12)Since ε is arbitrary, (7.11) finally yields ¯v(x) ≥ v(x).We note that this proof crucially relies on the fact that the control defined by(7.12) still belongs to A, being a measurable control. The possibility of obtaining

✐✐7.1. Optimal control problems: first examples 195an admissible control by joining together two different measurable controls is knownas concatenation property.The DPP can be used to characterize the value function in terms of a nonlinearpartial differential equation. In fact, let α ∗ ∈ A be the optimal control, and y ∗ theassociated evolution (to simplify, we are assuming that the infimum is a minimum).Then,that is,v(x) =∫ τ0g(y ∗ (s), α ∗ (s))e −λs ds + e −λτ v(y ∗ (τ)), (7.13)v(x) − e −λτ v(y ∗ (τ)) =∫ τ0g(y ∗ (s), α ∗ (s))e −λs ds (7.14)so that adding and subtracting e −λτ v(x) and dividing by τ, we gete −λτ (v(x) − v(y∗ (τ)))τ+ v(x)(1 − e−λτ )τ= 1 τ∫ τ0g(y ∗ (s), α ∗ (s))e −λs ds. (7.15)Assume now that v is regular. By passing to the limit for τ → 0 + , we havelim −v (y∗ (τ)) − v(x)= −Dv(x) · ẏ ∗ (x) = −Dv(x) · f(x, α ∗ (0))τ→0 + τlim v(x)(1 − e−λτ )= λv(x)τ→0 + τ∫1 τlim g(y ∗ (s), α ∗ (s))e −λs ds = g(x, α ∗ (0))τ→0 + τThen, we can conclude0λv(x) − Dv(x) · f(x, a ∗ ) − g(x, a ∗ ) = 0where a ∗ = α ∗ (0). Rewriting the DPP in the equivalent form{ ∫ τ}v(x) + supα∈A− g(y(s), α(s))e −λs ds − e −λτ v(y(τ))0= 0,we obtain the Hamilton–Jacobi–Bellman equation (or Dynamic Programming equation)corresponding to the infinite horizon problem,λu(x) + sup {−f(x, a) · Du(x) − g(x, a)} = 0. (7.16)a∈ANote that, given x, the value of a achieving the max (assuming it exists) correspondsto the control a ∗ = α ∗ (0), and this makes it natural to interpret the argmax in (7.16)as the optimal feedback at x. We will get back to this point in the sequel.In short, (7.16) can be written aswith x ∈ R d , andH(x, u, Du) = 0 (7.17)H(x, u, p) = λu(x) + sup {−f(x, a) · p − g(x, a)} . (7.18)a∈ANote that H(x, u, ·) is convex (being the sup of a family of linear functions) andthat, H(x, ·, p) is monotone (since λ > 0), so that we are in the framework of theexistence and uniqueness results presented in Chapter 2 (see Theorem 2.13).

✐✐196 Chapter 7. Control and Games7.1.2 Riccati equation and feedback controls for theLinear-Quadratic Regulator problemIn this problem, a linear (or linearized) controlled dynamical system must be stabilizedminimizing a linear combination of respectively the state and the controlenergy. The dynamics is given by:{ẏ(s) = Ay(s) + Bα(s)(7.19)y(0) = x,where A ∈ R d×d and B ∈ R d×m , x, y(s) ∈ R d and α(s) ∈ R m . The cost functionalis defined asJ x (α) = 1 2∫ +∞0(y(s; α) t Qy(s; α) + α(s) t Rα(s) ) e −λs ds (7.20)where Q ∈ R d×d and R ∈ R m×m are symmetric, strictly positive definite matrices,and λ ≥ 0. We are not interested in giving the most general framework of theproblem, but simply remark that it should be assumed that the system (7.19) iscontrollable, or that λ is large enough.We show now that this special structure of the problem allows to derive an explicitexpression for the optimal control in feedback form, as α(s) = Ky(s) where K isan m × d matrix defined via the solution of the so-called Riccati equation.To write the Bellman equation of the Linear-Quadratic Regulator (LQR), weassume that the value function has a quadratic structure:v(x) = 1 2 xt P x,so that we also have Dv(x) = P x, for some matrix P ∈ R d×d to be determined.With this choice, (7.16) takes the form{λ2 xt P x + max −(Ax + Ba) t P x − 1a2 xt Qx − 1 }2 at Ra = 0, (7.21)in which, given x, the argmax can be explicitly computed asUsing (7.22) into (7.21), we geta ∗ = −R −1 B t P x. (7.22)λ2 xt P x − x t A t P x − 1 2 xt Qx + 1 2 xt P t BR −1 B t P x = 0,that is,1 [( 2 xt λI − 2A t) P − Q + P t BR −1 B t P ] x = 0, (7.23)for any x ∈ R d . Therefore, the unknown matrix P should solve the quadratic RiccatiequationP t BR −1 B t P + ( λI − 2A t) P − Q = 0.Once computed the matrix P , the optimal control is obtained as a function of thecurrent state y(s) by means of (7.22), and more preciselyα ∗ (s) = −R −1 B t P y ∗ (s). (7.24)

✐✐7.2. Dynamic Programming for other classical control problems 197To check that (7.24) is indeed the optimal control, define the functionG(s; α) = e−λs2(y(s) t Qy(s) + α(s) t Rα(s) ) + e −λs y(s) t P (Ay(s) + Bα(s)) −− λe−λsy(s) t P y(s).2Now, substituting the expression (7.24) for α ∗ , and using the Riccati equation andsome elementary algebra, we can show thatOn the other hand, we also haveG(s; α) = e−λs2G(s; α ∗ ) = 0, for any s ∈ R + . (7.25)(y(s) t Qy(s) + α(s) t Rα(s) ) + d ds[e −λs v(y(s)) ] ,so that, using the control α ∗ and integrating on [0, T ], we get∫1 T(y ∗ (s) t Qy ∗ (s) + α ∗ (s) t Rα ∗ (s) ) e −λs ds + e −λT v(y ∗ (T )) − v(x).2 0Passing to the limit for T → ∞ and taking into account (7.25), we obtain that, inthe class of controls for which |y(s)| 2 = o(e λs ),12∫ ∞0(y ∗ (s) t Qy ∗ (s) + α ∗ (s) t Rα ∗ (s) ) e −λs ds = v(x),that is, the pair (y ∗ , α ∗ ) is optimal.The results above show the relationship between the max operator appearingin the Bellman equation, and the feedback form of the optimal control. Theyalso explain the popularity of the Linear-Quadratic Regulator problem in controlliterature: the optimal feedback control can be obtained via the solution of a matrixequation and has an explicit form. Actually, in many real problems of stabilization,the first try is to linearize the dynamics and approximate the cost by quadraticterms in order to directly apply the solution of the Riccati equation to the linearizedproblem.7.2 Dynamic Programming for other classical controlproblemsGiven a controlled dynamical system of the form (7.1)–(7.2), it is possible to considerother optimal control problems with different forms of the cost functional.We briefly review other classical problems and give their corresponding Bellmanequations.7.2.1 The finite horizon problemAssume the control system has the form (7.2), with t 0 = t, x 0 = x. In the Bolzaformulation, the functional of the finite horizon problem has the formJ x,t (α) =∫ Ttg(y x,t (s; α), α(s))e −λs ds + e −λ(T −t) ψ(y x,t (T ; α)) (7.26)

✐✐198 Chapter 7. Control and Gamesin which t ∈ [0, T ) and λ ≥ 0. For the finite horizon problem we can follow thesame arguments already seen for the infinite horizon problem. The first step is thedefinition of the value functionv(x, t) = infα∈A J x,t(α) (7.27)which in turn can be proved to satisfy the following DPP:Proposition 7.3 (DPP for the finite horizon problem). Under the assumptionsof Theorem 7.1, for all x ∈ R d and t < τ ≤ T ,{∫ τv(x, t) = infα∈A t}g(y x,t (s; α), α(s))e −λs ds + e −λ(τ−t) v(y x,t (τ; α), τ) (7.28)Then, in a similar way, it is possible to derive the Bellman equation in theform of an evolutive Hamilton–Jacobi equation with a terminal condition:⎧⎨−u t + λu + sup{−f(x, a) · Du − g(x, a)} = 0 (x, t) ∈ R d × (0, T ),a∈A(7.29)⎩u(x, T ) = ψ(x) x ∈ R d .We refer the interested reader to [E98] p. 557 for a complete proof.7.2.2 The optimal stopping problemIn this problem, beside the usual control α, it is possible to stop the dynamics atany intermediate time θ ∈ [0, +∞] (respectively, θ ∈ [t, T ] in the finite horizon case),paying a corresponding final cost given by ψ(y(θ)) (respectively, ψ(y(T ∧ θ))). Inthe infinite horizon case, the cost functional defining the optimal stopping problemis therefore given byJ x (θ, α) =∫ θ0g(y x (s; α), α(s))e −λs ds + e −λθ ψ(y(θ; α)). (7.30)Then, for x ∈ R d , θ ∈ [0, +∞] and α ∈ A, the value function is defined asv(x) = infθ,α J x(θ, α). (7.31)This problem can be rephrased as a standard infinite horizon (respectively, finitehorizon) problem by adding to the control set A a new control ā, such that f(y, ā) =0 and g(y, ā) ≡ 0 for any y ∈ R d . A more explicit way of treating the problem is toderive a DPP which takes into account the option of stopping at a finite time, thenobtain the Bellman equation, which has the form of an obstacle problem:max(H(x, u, Du), u − ψ) = 0, (7.32)where the Hamiltonian function H is defined as in (7.16). In (7.32), the set on whichu = ψ corresponds to the so-called stopping set, in which the optimal strategy isprecisely to stop the system.

✐✐7.2. Dynamic Programming for other classical control problems 1997.2.3 The minimum time problemAnother classical example is the minimum time problem. We assume a dynamicsin the form (7.2), which has to be steered in the shortest possible time to a targetset T ⊂ R d . This set is given and we will always assume thatThe target set T is closed and int(T ) ≠ ∅, ∂T is sufficiently regular. (7.33)For every control α ∈ A we can define the first time of arrival on the target T as{t x (α) = inf s ∈ R + : y x (s; α) ∈ T }sprovided the set at the right-hand side in nonempty. If the set is empty, that is, ify x (s; α) /∈ T for any s > 0, it is natural to define t x (α) = +∞. This leads to thefollowing definition:{+∞ if yx (s; α) /∈ T ∀s > 0t x (α) = {inf s ∈ R + : y x (s; α) ∈ T } (7.34)elsesSince we want to minimize the time of arrival on the target, the value function forthis problem is given byT (x) = inf t x(α) (7.35)α∈AA basic example, discussed in Chapter 1, is the case in whichẏ(s) = α(s)with α(s) ∈ B(0, 1). In this case, the minimum time T (x) equals the distanced(x, T ) and can be a nonsmooth function whenever T is nonconvex.Dynamic Programming for the minimum time problem Consider a system drivenby (7.2). Since the the minimum time function (i.e. the value function for thisproblem) may not be finite everywhere, we need to define its domain.Definition 7.4 (Reachable set). The reachable set is defined as the set of initialstates from which it is possible to reach the target, that is,R := { x ∈ R d : T (x) < +∞ } .The reachable set clearly depends on the target, on the dynamics and on theset of admissible controls. However, this dependence is complex even in simplesituations, so that the reachable set cannot be regarded as a datum in our problem.This means that we have to determine the pair (T, R) as the solution of a freeboundary problem.In order to derive the HJB equation for the minimum time problem, we state therelated Dynamic Programming Principle.Proposition 7.5 (DPP for the minimum time problem). For all x ∈ R andτ ∈ [0, T (x)) (so that x /∈ T ),T (x) = infα∈A {τ + T (y x(τ; α))}. (7.36)

✐✐200 Chapter 7. Control and GamesThe proof of this DPP relies basically on the same arguments used for theinfinite horizon case, and uses the possibility of generating new admissible controlsby concatenation and/or translation. To derive the Hamilton–Jacobi–Bellmanequation from the DPP, we rewrite (7.36) for t ∈ (0, T (x)) asand divide by τ, obtainingsupα∈AT (x) − infα∈A T (y x(τ; α)) = τ{ }T (x) − T (yx (τ; α))= 1.τIn order to pass to the limit as τ → 0 + we assume, for the moment, that T isdifferentiable at x and that lim τ→0 + commute with sup α . Provided ẏ x (0; a) exists,we getsup {−DT (x) · ẏ x (0, α)} = 1,α∈Aso that, if lim τ→0 + α(τ) = a, we obtainsup {−DT (x) · f(x, a)} = 1, (7.37)a∈Awhich is the Hamilton–Jacobi–Bellman equation associated to the minimum timeproblem. This derivation of the HJB equation basically shows that if T is regular,then it is a classical solution of (7.37), i.e., proves the following proposition:Proposition 7.6. If T ∈ C 1 in a neighborhood of x ∈ R\T , then T satisfies (7.37)at x.However, (7.37) needs to be complemented with a boundary condition, the“natural” condition beingTherefore, once defined the HamiltonianT (x) = 0 (x ∈ ∂T )H(x, p) := sup{−p · f(x, a)} − 1,a∈Awe can rewrite (7.37) as a boundary value problem in the compact form{H(x, DT (x)) = 0 x ∈ R \ T ,T (x) = 0 x ∈ ∂T .(7.38)Note again that H(x, ·) is convex since it is the sup of linear operators.We have already seen that T may not be differentiable even in the simplestcases. In addition, the continuity of T around the target may also fail to hold,unless the following property of Small-Time Local Controllability (STLC) aroundT is assumed.Definition 7.7 (Small Time Local Controllability). Assume ∂T is smooth.Then, the property of STLC is satisfied if, for any x ∈ ∂T , there exists a controlvector â ∈ A such that:f(x, â) · η(x) < 0,

✐✐7.2. Dynamic Programming for other classical control problems 201where η(x) is the exterior normal to ∂T at x.The STLC ensures that R is an open subset of R d and that T is continuousup to ∂T . Moreover, for all z ∈ ∂R,lim T (x) = +∞.x→zWith this last assumption, it is possible to prove that T ( · ) is a viscosity solutionof the boundary value problem (7.38).Theorem 7.8. If R \ T is open and T is continuous, then T is a viscosity solutionof (7.38).Proof. For τ small enough, by the DPP we have that, for any ᾱ ∈ A:and that, given ε > 0, there exists an α ε ∈ A such that:T (x) − T (ȳ(τ)) ≤ τ (7.39)T (x) − T (y ε (τ)) ≥ t(1 − ε), (7.40)where, as usual, ȳ(τ) = y x (τ; ᾱ) and y ε (τ) = y x (τ; α ε ).Consider now a test function φ ∈ C 1 (R d \ T ) such that T − φ has a local maximumat x. Then, for τ small enough, we haveand by (7.39):T (x) − φ(x) ≥ T (ȳ(τ)) − φ(ȳ(τ)),φ(x) − φ(ȳ(τ)) ≤ T (x) − T (ȳ(τ)) ≤ τ,so that, dividing by τ and passing to the limit for τ → 0 + (as it has been done inthe derivation of (7.37)), we getH(x, ∇φ(x)) ≤ 0.Similarly, if φ ∈ C 1 and T − φ has a local minimum at x, for τ small enough wehaveT (x) − φ(x) ≤ T (y ε (τ)) − φ(y ε (τ)),and by (7.40):φ(x) − φ(y ε (τ)) ≥ T (x) − T (y ε (τ)) ≥ τ(1 − ε).Since ε is arbitrary, dividing by τ and passing to the limit, we obtainH(x, ∇φ(x)) ≥ 0,which fulfills the definition of viscosity solution.Using STLC property, it is also possible to prove that the boundary condition issatisfied on ∂T . This part of the proof will be skipped.A different approach to the solution of (7.37) is to perform a suitable rescalingof T , obtained via the auxiliary function{1 if T (x) = +∞v(x) :=1 − e −µT (x) (7.41)else,

✐✐202 Chapter 7. Control and Gamesfor some positive µ. It is easy to check that v is itself a value function, that is,where J x is given byv(x) = infα∈A J x(α),J x (α) =∫ tx(α)0e −µs dsand t x (α) is defined by (7.34). This control problem is nothing but an infinitehorizon problem (with constant running cost g(x, a) ≡ 1) in which the state of thesystem is stopped as soon as it reaches the target T , with zero stopping cost. TheDynamic Programming equation is in the form of a boundary value problem:⎧⎨µv(x) + sup{−Dv · f(x, a) − 1} = 0 x ∈ R d \ Ta∈A(7.42)⎩v(x) = 0 x ∈ ∂T .The change of variable (7.41) is called Kružkov transformation and gives severaladvantages. First, v(x) takes values in [0, 1] (whereas T is generally unbounded)and this helps both in the analysis and in the numerical approximation. Second,the formulation (7.42) avoids any reference to the reachable set. Once obtained v,the minimum time T and reachable set R can be recovered by the relationshipsT (x) = −ln(1 − v(x)), R = {x ∈ R d : v(x) < 1}.µIn addition, the formulation as an infinite horizon problem allows the time-marchingnumerical scheme to be a contraction mapping. We will come back to this point inthe section on numerical schemes.7.3 The link between Bellman equation andPontryagin Maximum PrincipleThere exists an important link between the DPP and the Pontryagin MaximumPrinciple (PMP in the sequel), which gives a set of necessary conditions to characterizethe optimal open-loop solution (y ∗ (·), α ∗ (·)). Using this relationship, wederive here the PMP for the finite horizon problem (7.1), (7.26) under additionalsimplifying assumptions on the data. In particular, we will assume that ψ ∈ C 2 (R d ),that Df is continuous, and, above all, that v ∈ C 2 (R d ) so that the Hamilton–Jacobiequation (7.29) is verified in a classical sense. We also assume, for simplicity, thatg ≡ 0 and λ = 0 so that, by the DPP, the functionh(s) := v(y(s; α), s)is nondecreasing for every α, and is constant only for the optimal trajectory. Then,an optimality condition for a pair (y ∗ , α ∗ ) is that, for any s ∈ [0, T ]:h ′ (s) = f(y ∗ (s), α ∗ (s)) · Dv(y ∗ (s), s) + v s (y ∗ (s), s) = 0. (7.43)On the other hand, since at any point (z, s) ∈ R d × [0, T ] the Hamilton–Jacobi–Bellman equation gives−v t (z, s) − f(z, a) · Dv(z, s) ≤ 0 for every a ∈ A, (7.44)

✐✐7.3. The link between Bellman equation and Pontryagin Maximum Principle 203setting z = y ∗ (s), by (7.43)–(7.44) we obtain an equivalent condition known as theMinimum Principle:f(y ∗ (s), α ∗ (s)) · Dv(y ∗ (s), s) = mina∈A f(y∗ (s), a) · Dv(y ∗ (s), s). (7.45)Now, in order to prove the PMP, we recall that, by (7.43), the left-hand side of (7.44)vanishes at (z, s) = (y ∗ (s), s) for a = α ∗ (s) if the control α ∗ is optimal. This impliesthat the trajectory y ∗ (s) maximizes the map z → −v t (z, s) − Dv(z, s) · f(z, α ∗ (s))and that, due to our regularity assumptions, we can differentiate this map withrespect to z obtaining−Dv t − J f Dv − D 2 vf = 0, (7.46)in which J f is the d × d Jacobian matrix for f and D 2 v is the d × d Hessian matrixfor v. Let us now define the co-state or adjoint vector p(t) as the vectorp(s) := Dv(y(s)), s). (7.47)Differentiating with respect to s we obtain, by (7.46) and the regularity of v,p ′ = D 2 vf − Dv t = −J f p. (7.48)This means that the co-state associated with (y(s), s) and α ∗ can be redefined,without making reference to the value function, as the unique solution on [0, T ] ofthe system of linear differential equationswith the terminal conditionp ′ (s) = −J f (y(s), α(s))p(s), (7.49)p(T ) = Dψ(y(T )) (7.50)The necessary optimality conditions on the couple (y ∗ (t), α ∗ (t)) are then written as−p ∗ (t) · f(y ∗ (t), α ∗ (t)) = max{−p(t) · f(y(t), a)} (7.51)a∈Av t (y ∗ (t), T − t) = p ∗ (t) · f(y ∗ (t), α ∗ (t)) = −H(y ∗ (t), p ∗ (t)) ?? (7.52)In a remarkably technical way, Pontryagin’s Maximum Principle can also be derivedin a more general setting which, in particular, does not require the regularityassumptions on the data and on the value function.Theorem 7.9. Assume v ∈ C 2 (R d ), A compact, ψ ∈ C 2 (R d ), f differentiable withrespect to x and Df continuous, Let y(·) = y x0 (·; α) and let the co-state p(·) be thecorresponding solution of (7.49)-(7.50). Then, the open-loop control α ∗ is optimalfor (x, T ) if and only if for almost any t ∈ (0, T ]:−p ∗ (t) · f(y ∗ (t), α ∗ (t)) = max{−p(t) · f(y(t), a)} := H(y(t), p(t))a∈A(p ∗ (t), H(y ∗ (t), p ∗ (t))) ∈ D + v(y(t), T − t)We refer the interested reader to [BCD97] for a more general proof of thisresult.

✐✐204 Chapter 7. Control and Games7.4 SL schemes for optimal control problemsThe construction of SL schemes for Dynamic Programming equations follows thesame guidelines shown in Chapters 5–6 for the simpler model problems. In fact,equations (5.70) or (5.128) may be seen as special cases of HJB equations for controlproblems in which⎧⎪⎨ f(x, a) = a,g(x, a) = H⎪⎩∗ (a),(7.53)ψ(x) = u 0 (x)(with the minor difference that (5.70) is a forward, instead of a backward, equation).When considering more general DP equations, it should be clear that optimaltrajectories play the role of characteristics, but they are no longer expected to bestraight lines. Therefore, in a sense, time discretization must combine strategiesused in both advection and convex HJ equations.For all these reasons, as well as for the sake of obtaining explicit error estimates,we are led to consider time discretization as an intermediate (semi-)discretizationstep. This step also has the meaning of approximating the original controlproblem with a problem in discrete time, and is in fact the situation in which theDP Principle has been first formulated and applied.Next, the space discretization is performed as in Chapter 5, unless for replacing theHopf–Lax formula with the discrete version of the Dynamic Programming Principle.This section will review the construction and convergence analysis of SLschemes for HJB equations, with a special emphasis on the control problems examinedin Sections 7.1–7.2. For simplicity, we will consider schemes on the whole ofR d , and low-order (monotone) implementations. The treatment of boundary conditions,along with the construction of high-order versions, can be performed withthe techniques discussed in Chapters 5–6.7.4.1 Approximation of the infinite horizon problemWe sketch the basic ideas in the case of the infinite horizon problem. The plan isto construct a time-marching scheme, starting with time discretization.Time discretization In order to give a discrete time approximation of the controlproblem, we fix a time step ∆t and set t m = m∆t (m ∈ N). The simplest wayto discretize (7.2) is by using the explicit Euler scheme, which corresponds to thefollowing discrete dynamical system:{y m+1 = y m + ∆tf(y m , a m )(7.54)y 0 = x.Here, the sequence of vectors a m ∈ A has the role of a (discrete) control. We willdenote by α ∆t = {a m } the sequence as a whole, and possibly identify this sequence(with a slight abuse of notation) with the continuous, piecewise constant controldefined byα ∆t (s) = a m , s ∈ [t m , t m + 1). (7.55)Whenever we will need to emphasize the dependence of the trajectory on x and α ∆twe will use the notation y m (x; α ∆t ).

✐✐7.4. SL schemes for optimal control problems 205Remark 7.10. It is also possible to implement higher order time discretizationsby using, e.g., Runge–Kutta type schemes in (7.54). We refer to [FF94] for a moreextensive study of this technique, whereas an example of application will be given inthe numerical tests section.Given the time discretization (7.54) for the controlled system, the correspondingdiscrete version of the cost functional J x may be obtained by a rectangle quadrature:Jx∆t (α∆t ) +∞∑:= ∆t g(y m , a m )e −λtm , (7.56)m=0in which the dependence on α ∆t also appears via the trajectory points y m .The discrete dynamics (7.54) and cost functional (7.56) define a discrete time controlproblem, whose value function isv ∆t (x) := inf J ∆tα ∆t x(α∆t ) . (7.57)Adapting the arguments of the continuous case, it is possible to prove a DiscreteDPP for the optimal control problem (7.54)–(7.56):Proposition 7.11 (DDPP for the infinite horizon problem). Fix ∆t > 0.Then, for all x ∈ R d and any positive integer ¯n:{}∑¯n−1v ∆t (x) = inf ∆t g(y m , a m )e −λtm + e −λt¯n v ∆t (y¯n ) . (7.58)α ∆t m=0The discrete time version of the SL scheme can be obtained by choosing ¯n = 1in (7.58). This gives an approximation in the fixed point formv ∆t {(x) = min ∆tg(x, a) + e −λ∆t v ∆t (x + ∆tf(x, a)) } (7.59)a∈Ain which e −λ∆t is sometimes replaced by its first order Taylor approximation 1−λ∆t(this has no influence on the consistency rate, but introduces a constraint on ∆t toget a contraction in the right-hand side of (7.59)).Space discretization The space discretization of the semi-discrete approximation(7.59) is performed in a completely standard way by replacing the value v ∆t (x +∆f(x, a)) by a P 1 or Q 1 interpolation. We end up with the fully discrete scheme{v j = min ∆tg(xj , a) + e −λ∆t I 1 [V ](x j + ∆tf(x j , a)) } . (7.60)a∈ADenoting the right-hand side of (7.60) as S(∆, V ), the scheme may be written asos also, in iterative form, asV = S(∆, V )V (k+1) = S(∆, V (k) ) (7.61)where k is the iteration index. An analysis of the scheme along the guidelines ofChapter 5 shows that:

✐✐206 Chapter 7. Control and Games• S(∆, ·) is monotone;• S(∆, ·) is a contraction in l ∞ , and therefore V (k) converges towards a uniquesolution V ∈ l ∞ for any initial guess V (0) . The contraction coefficient is easilyshown to be L S = e −λ∆t and hence, as usual in time-marching schemes, convergencemay become very slow as ∆t → 0. Suitable acceleration techniques(exploiting monotonicity of the operator S) have been constructed to handlethis problem (see [F87, F97] for more details).• The scheme is l ∞ stable if the running cost g is bounded. In fact, since theinterpolation I 1 is nonexpansive, from (7.60) we have, assuming that V (0) = 0:∥ ∥ ∥∥V (k+1) ∥∥∞≤ ∆t‖g‖ ∞ + e −λ∆t ∥∥V (k)∥ ≤∞≤ · · · ≤ ∆t‖g‖ ∞(1 + e −λ∆t + e −2λ∆t + · · · ) ≤≤∆t‖g‖ ∞1 − e −λ∆t = ‖g‖ ∞(1 + O(∆t))λConvergence analysis First, we point out that a first convergence analysis mightbe performed by means of Barles–Souganidis theorem, the scheme being monotone.When passing from the Hopf–Lax formula to the Dynamic Programming Principle,however, consistency analysis (if carried out with the arguments of Chapter 5)requires more caution.Rather than duplicating the analysis performed in Chapter 5, we will applyhere the same arguments to estimate the rate of convergence of the approximatesolution to the value function, i.e., to the solution of the Bellman equation (7.16).At this level, it can be convenient to separate time from space discretization.Concerning time discretization, we first assume that v is Lipschitz continuous,and in addition, that(i) ‖f‖ is uniformly bounded. Moreover, f(x, a) is linear with respect to a, i.e.,it has the structuref(x, a) = f 1 (x) + F 2 (x)awith f 1 : R d → R d and F 2 (x) is a d × m matrix;(ii) g(·, a) is Lipschitz continuous, and g(x, ·) is convex;(iii) There exists an optimal control α ∗ ∈ A (this could be derived from the twoassumptions above, but we state it explicitly).First, making explicit the minimum in (7.7) and (7.59), we can rewrite v andv ∆t asv(x) =∫ ∆t0g(y x (s; α ∗ ), α ∗ (s))e −λs ds + e −λ∆t v(y x (∆t; α ∗ )),v ∆t (x) = ∆tg(x, a ∗ ) + e −λ∆t v ∆t (x + ∆tf(x, a ∗ )).Then, we can bound v ∆t (x) − v(x) from above by using in the continuous problemthe constant control α(s) ≡ a ∗ (which is suboptimal), obtainingv(x) − v ∆t (x) ≤∫ ∆t0g(y x (s; a ∗ ), a ∗ )e −λs ds + e −λ∆t v(y x (∆t; a ∗ )) −−∆tg(x, a ∗ ) − e −λ∆t v ∆t (x + ∆tf(x, a ∗ )).

✐✐7.4. SL schemes for optimal control problems 207Now, by elementary approximation arguments,whereas∫ ∆t0g(y x (s; a ∗ ), a ∗ )e −λs ds − ∆tg(x, a ∗ ) = O ( ∆t 2) ,v(y x (∆t; a ∗ )) − v ∆t (x + ∆tf(x, a ∗ )) ≤ v(y x (∆t; a ∗ )) − v(x + ∆tf(x, a ∗ )) ++v(x + ∆tf(x, a ∗ )) − v ∆t (x + ∆tf(x, a ∗ )) ≤≤ O ( ∆t 2) + ∥ ∥ v − v∆t ∥ ∥∞,where the first term follows from the Lipschitz continuity of v and elementary resultsfor the Euler scheme. We can therefore deduce thatv(x) − v ∆t (x) ≤ O ( ∆t 2) + e −λ∆t ∥ ∥ v − v∆t ∥ ∥∞. (7.62)Note that this unilateral estimation does not require assumptions (i) and (ii) above.To perform the reverse estimate, we should replace the discrete optimal control a ∗by a suboptimal ā, so thatv ∆t (x) − v(x) ≤ ∆tg(x, ā) + e −λ∆t v ∆t (x + ∆tf(x, ā)) − (7.63)−∫ ∆t0g(y x (s; α ∗ ), α ∗ (s))e −λs ds − e −λ∆t v(y x (∆t; α ∗ )).A suitable choice for ā is to define it as the integral mean of α ∗ over [0, ∆t]:ā = 1 ∆t∫ ∆t0α ∗ (s)ds.In fact, with this choice we can write y ∗ (∆t) = y x (∆t; α ∗ ) asy ∗ (∆t) = x += x +On the other hand,∫ ∆t0∫ ∆t0∫ ∆t0f 1 (y ∗ (s))ds +∫ ∆t[f 1 (x) + O(∆t)]ds += x + ∆tf 1 (x) + F 2 (x)0∫ ∆t0F 2 (y ∗ (s))α ∗ (s)ds =∫ ∆t= x + ∆t[f 1 (x) + F 2 (x)ā] + O ( ∆t 2) =0[F 2 (x) + O(∆t)]α ∗ (s)ds =α ∗ (s)ds + O ( ∆t 2) == x + ∆tf(x, ā) + O ( ∆t 2) . (7.64)g(y ∗ (s), α ∗ (s))e −λs ds ==∫ ∆t0∫ ∆t0[g(x, α ∗ (s)) + O(∆t)](1 + O(∆t))ds =g(x, α ∗ (s))ds + O ( ∆t 2) ≥≥ ∆tg(x, ā) + O ( ∆t 2) , (7.65)where the last inequality follows from Jensen’s inequality, due to the convexity ofg(x, ·). Using (7.64)–(7.65) in (7.63), and retracing the proof of (7.62), we getv ∆t (x) − v(x) ≤ O ( ∆t 2) + e −λ∆t ∥ ∥ v − v∆t ∥ ∥∞, (7.66)

✐✐208 Chapter 7. Control and Gameswhich gives, taking into account (7.62) itself:∣ v ∆t (x) − v(x) ∣ ∣ ≤ O(∆t2 ) + e −λ∆t ∥ ∥ v − v∆t ∥ ∥∞. (7.67)Due to the assumptions on f and g, the O(∆t 2 ) term is uniform with respect to x,so we can pass to the ∞-norm in (7.67), obtaining that, for some positive constantC: (1 − e−λ∆t ) ∥ ∥v ∆t − v ∥ ∥∞≤ C∆t 2 ,and therefore ∥ ∥v ∆t − v ∥ ∥∞≤ C∆t. (7.68)This estimate has been proved (with different techniques) in [CDI84], under theassumption that v is semiconcave. If v is only Lipschitz continuous, and assumptions(i)–(iii) above are dropped, then∥ v ∆t − v ∥ ∥∞≤ C∆t 1/2 . (7.69)Note that a sufficient condition for the Lipschitz continuity of v is to have Lipschitzcontinuous data (f and g) and λ > L f .For the fully discrete scheme (7.60), we estimate the space discretization error comparingwith the same technique above v j and v ∆t (x j ). Using the control a ∗ , optimalfor v ∆t , and denoting z ∗ j = x j + ∆tf(x j , a ∗ ), we obtain the one-sided estimate:v j − v ∆t (x j ) ≤ ∆tg(x j , a ∗ ) + e −λ∆t I 1 [V ](z ∗ j )) −−∆tg(x j , a ∗ ) + e −λ∆t v ∆t (z ∗ j )) ≤≤ e −λ∆t ∣ ∣ I1 [V ](z ∗ j ) − I 1[V∆t ] (z ∗ j ) ∣ ∣ ++e −λ∆t ∣ ∣ I1[V∆t ] (z ∗ j ) − v ∆t (z ∗ j ) ∣ ∣ ≤≤ e −λ∆t ∥ ∥V − V ∆t∥ ∥∞+ O(∆x),where V ∆t is the vector of samples of v ∆t and, by the Lipschitz continuity of v ∆t ,we have bounded its interpolation error with O(∆x). Moreover, we have used themonotonicity of the reconstruction I 1 to bound the error at zj∗ with the error onthe nodes.Giving the opposite bound and passing to the ∞-norm, we obtain∥ V ∆t − V ∥ ∞≤ C ∆x∆t . (7.70)Combining time and space discretization error estimates, we have the completeestimate:|v j − v(x j )| ≤ C 1 ∆t γ ∆x+ C 2 (7.71)∆twhere γ = 1/2 or γ = 1, depending on the assumptions. In the former case, thebest coupling is obtained for ∆x = ∆t 3/2 , in the latter for ∆x = ∆t 2 .7.4.2 Approximation of the finite horizon problemLet us briefly review the results (proved in [FG98]) for the finite horizon problem.Time discretization uses the time step ∆t = T/N, the time horizon of the problemis [t n , t N ] = [n∆t, T ] and the discrete dynamical system is defined by{y m+1 = y m + ∆tf(y m , a m )(7.72)y n = x,

✐✐7.4. SL schemes for optimal control problems 209whereas the cost functional is discretized byJx,t ∆t (nα∆t ) N−1∑:= ∆t g(y m , a m )e −λtm + e −λ(t N −t n) ψ(y N ), (7.73)m=nin which again the dependence on the discrete control appears also through thepoints y m . The value function for the discrete time problem is naturally defined asand satisfies the corresponding DDPP:v ∆t,n (x) := infα ∆t J ∆tx,t n(α∆t ) (7.74)Proposition 7.12 (Discrete DPP for the finite horizon problem).∆t = T/N. Then, for all x ∈ R d , ¯n ∈ {n + 1, . . . , N},{}∑¯n−1v ∆t,n (x) = inf ∆t g(y m , a m )e −λtm + e −λ(t¯n−tn) v ∆t,¯n (y¯n ) . (7.75)α ∆t m=nSetThe discrete time version of the SL scheme is obtained setting ¯n = n + 1 in(7.75) and enforcing the final condition at time T , that is{v ∆t,n (x) = min a∈A{∆tg(x, a) + e −λ∆t v ∆t,n+1 (x + ∆tf(x, a)) }v ∆t,N (x) = ψ(x),(7.76)so that, as for the continuous problem, the solution is obtained backward in timefrom t N = T to t 0 = 0. Last, a P 1 or Q 1 interpolation is applied to reconstruct thevalue v ∆t,n+1 at the right-hand side of (7.76), giving the fully discrete scheme{ {vnj = min ∆tg(xj , a) + e −λ∆t [I 1 Vn+1 ] (x j + ∆tf(x j , a)) } ,a∈Avj N = ψ(x j ).(7.77)Convergence analysis We outline the main results of convergence for the finitehorizon problem. First, the scheme is monotone and consistent, and hence convergentby the Barles–Souganidis theorem. Concerning error bounds, if the solutionv of the Bellman equation (7.28) is Lipschitz continuous, then the discrete timeapproximation satisfy the estimate‖v ∆t − v‖ ∞ ≤ C∆t 1/2 (7.78)where C is a positive constant which in general depends on T . Note that Lipschitzcontinuity of the data f and g imply again Lipschitz continuity of the value function.For the fully discrete scheme (7.77) we have (see [FG98]):|vj n − v(x j , t n )| ≤ C 1 ∆t 1/2 ∆x+ C 2 , (7.79)∆t1/2 and in this case the best coupling is obtained by choosing ∆x = ∆t.

✐✐210 Chapter 7. Control and Games7.4.3 Approximation of the optimal stopping problemIn the (infinite horizon) optimal stopping problem, it is possible to use the argumentsgiven in the continuous formulation to derive the time discrete approximation(v ∆t {(x) = min ∆tg(x, a) + e −λ∆t v ∆t (x + ∆tf(x, a)) } ), ψ(x) . (7.80)mina∈AIn (7.80) it is possible to recognize that, in addition to the continuous control, thecontroller has the further choice of stopping the system, in which case a price ψis paid. The optimal strategy clearly corresponds to the lower payoff, this beingselected by the outer min operator.In the fully discrete version, the value of v ∆t is replaced as usual with a P 1 or Q 1interpolation, and therefore the scheme is written as()v j = minmina∈A{∆tg(xj , a) + e −λ∆t I 1 [V ](x j + ∆tf(x j , a)) } , ψ(x j ). (7.81)Denoting again by S(∆, V ) the right-hand side of (7.60), and by Ψ the vector ofsamples of the function ψ, the scheme can be put in the compact formwhich allows for the iterative versionV = min (S(∆, V ), Ψ) , (7.82)V (k+1) = minAn equivalent form of the scheme (7.82) is also()S(∆, V (k) ), Ψ .max (V − S(∆, V ), V − Ψ) = 0,which, dividing the first argument by ∆t, may be rewritten as( )V − S(∆, V )max, V − Ψ = 0.∆tIn this form, it is immediate to recognize that the scheme is consistent with (7.32).To check that (7.82) is monotone, assume that U ≥ W component by component.Then, the scheme is monotone ifNow, in the caseas well as in the casemin (S(∆, U), Ψ) − min (S(∆, W ), Ψ) ≥ 0. (7.83)min(S(∆, U), Ψ) = min(S(∆, W ), Ψ) = Ψ,min(S(∆, U), Ψ) = S(∆, U)min(S(∆, W ), Ψ) = S(∆, W ),condition (7.83) is clearly satisfied, S being monotone. On the other hand, ifmin(S(∆, U), Ψ) = S(∆, U)min(S(∆, W ), Ψ) = Ψ,

✐✐7.4. SL schemes for optimal control problems 211we can writeand henceS(∆, W ) ≥ min(S(∆, W ), Ψ) = Ψ,min (S(∆, U), Ψ) − min (S(∆, W ), Ψ) = S(∆, U) − Ψ ≥≥ S(∆, U) − S(∆, W ) ≥ 0so that (7.83) is satisfied again. Since the same argument proves the symmetriccase, we can conclude that the scheme is also monotone and therefore convergentby the Barles–Souganidis theorem.The finite horizon optimal stopping can be treated in a similar way.7.4.4 Approximation of the minimum time problemIn the previous section, we have shown how the Kružkov change of variable (7.41)allows to work on the (rescaled) value function as the unique viscosity solution of(7.42). This will be the standing line of work for the numerical approximation ofthe minimum time problem.Time discretization In order to introduce the time discrete minimum time problemfor the dynamics (7.54), we start by defining the time discrete analogue of thereachable set,R ∆t := { x ∈ R d : ∃α ∆t and m ∈ N such that y m(x; α∆t ) ∈ T } (7.84)and of the number of steps of first arrival,N ( {x, α ∆t) +∞ x /∈ R ∆t=(min{m ∈ N : y ) m x; α∆t∈ T } x ∈ R ∆t (7.85)The role of the minimum time is played now by the minimal number of stepsnecessary to reach the target starting at xN(x) := minα ∆t N ( x, α ∆t) , (7.86)so that the discrete approximation of the minimum time function T (x) is now∆tN(x). By an adaptation of the usual arguments, it is possible to prove thefollowingProposition 7.13 (DDPP for the minimum time problem). Let ∆t > 0 befixed. For all x ∈ R ∆t and 0 ≤ ¯n < N(x),N(x) = infα ∆t {¯n + N(yx(¯n; α∆t ))} . (7.87)However, rather then performing a time discretization on the basis of (7.87),we further apply the Kružkov change of variable (for simplicity, with µ = 1):v ∆t (x) = 1 − e −∆tN(x) . (7.88)

✐✐212 Chapter 7. Control and GamesNote that 0 ≤ v ∆t ≤ 1, as in the continuous case, and that v ∆t has constant valueon the set of initial points x which can be driven to T by the discrete dynamicalsystem in the same number of steps.Writing (7.87) for ¯n = 1, and changing variable, we characterize v ∆t as the solutionof{v ∆t {(x) = min e −∆t v ∆t (x + hf(x, a)) } + 1 − e −∆t x ∈ R d \ Ta∈A(7.89)v ∆t (x) = 0x ∈ Twhere the boundary condition on T stems from the definition of N(x, α ∆t ).Space discretization The space discretization of (7.89) is performed as above bya P 1 or Q 1 interpolation. The iterative fully discrete scheme reads⎧[⎨v (k+1)j = e −∆t min{I 1 V (k)] }(x j + hf(x j , a)) + 1 − e −∆t x j ∈ R d \ Ta∈A⎩v (k+1)j = 0 x j ∈ T(7.90)where k is the iteration index. Unless for the second condition, which works as aboundary condition, the scheme has the same structure of (7.61), so that denotingthe right-hand side of (7.90) as S(∆, V (k) ), it can easily be shown that:• S(∆, ·) is monotone;• S(∆, ·) is a contraction in l ∞ (with contraction coefficient L S = e −∆t ), andtherefore V (k) converges towards a unique solution V ∈ l ∞ for any initialguess V (0) . If V (0) is chosen as a numerical supersolution, then the sequenceis monotonically decreasing, and if in particularv (0)j ={0 x j ∈ T1 x j ∉ T ,then at the step k the scheme updates only the nodes which are driven to thetarget in precisely k steps. This lends itself to a fast implementation (thispoint will be further discussed in Chapter ??).Clearly, the presence of a boundary condition requires some extra effort intreating possibly complex geometries (e.g., by means of unstructured meshes asdiscussed in Chapter 3), as well as in implementing a variable step technique (asshown in Chapter 5). Moreover, when restricting to a bounded computationaldomain, a further, suitable boundary condition must be imposed on the externalboundary. We skip the problem here, and refer the reader to [F97] for a discussionof this point.Convergence analysis In order to prove an error bound for the discrete time approximationwe need to introduce a discrete analogue of the Small Time LocalControllability assumption. We define the δ-neighbourhood of ∂T asT δ := ∂T + δB(0, 1)(where, for shortness, d(x) = d(x, ∂T )). A first technical result states the followinglocal upper bound for the discrete time approximation around T :

✐✐7.4. SL schemes for optimal control problems 213Lemma 7.14. Let the assumptions of Theorem 7.1 be satisfied and let STLC holdtrue. Then, there exist positive constants ∆t, δ such that, for all ∆t < ∆t andx ∈ T δ :v ∆t (x) ≤ C d(x) + ∆t.We are now able to prove a convergence result:Theorem 7.15. Let the assumptions of Lemma 7.14 be satisfied and let T becompact with nonempty interior.Then, v ∆t converges to v locally uniformly in R d for ∆ → 0 +Proof. Since v ∆t is a discontinuous function, we define the two semicontinuousenvelopesv = lim inf v ∆t (y), v = lim sup v ∆t (y)h→0 +h→0y→x+y→xNote that v (respectively, v) is lower (respectively, upper) semicontinuous. The firststep is to show that1. v is a viscosity subsolution for (7.37)2. v is a viscosity supersolution for (7.37)Then, we want show that both the envelopes satisfy the boundary condition on T .In fact, by Lemma 7.14,v ∆t (x) ≤ Cd(x) + ∆twhich impliesTherefore, we have|v| ≤ Cd(x) (7.91)|v| ≤ Cd(x). (7.92)v = v = 0 on ∂T .Since the two envelopes coincide on ∂T we can apply the comparison theorem forsemicontinuous sub and supersolutions obtainingv = v = v on R d .For the discrete time approximation, it is also possible to give an error estimate,restricted to a compact set Ω ⊂ R in which the following condition issatisfied:∃ C 0 > 0 : ∀ x ∈ Ω there exists a time optimal control withtotal variation less than C 0 bringing the system to T .The typical choice for Ω is to take an hypercube containing the target set T .(7.93)Theorem 7.16. Let the assumptions of Lemma 7.14 be satisfied, and let Ω be acompact subset of R where assumption (7.93) holds. Then, there exists two positiveconstants ∆t and C such that, for all ∆t ≤ ∆t and x ∈ Ω,|v ∆t (x) − v(x)| ≤ C∆t. (7.94)

✐✐214 Chapter 7. Control and GamesThe above results show that the rate of convergence for the scheme based onthe Euler discretization is 1, which is exactly what we expected. A complete proofcan be found in [BF90b]. The analysis of the fully discrete scheme leads to ana-priori error estimate hat we will not prove here( ) ) 2 ∆x‖v j − v(x j )‖ ∞ ≤ C∆t(1 1/2 +(7.95)∆tA sketch of that proof will be given in the section on differential games, since for asingle player that problem reduces to the minimum time problem.7.4.5 Approximation of optimal feedback and trajectories forcontrol problemsOne of the main goals of numerical schemes for control problems is to computeapproximate optimal controls. As it has already been pointed out, the algorithmsproposed and analyzed in this chapter inherently compute an approximate optimalcontrol at every point of the grid. This information will not require extra effort.However, this is not sufficient to compute accurately the optimal feedback in thecomputational domain Ω and to derive the corresponding optimal trajectories. Thenumerical solution v, which is computed at the nodes, is extended to to the wholedomain by interpolation, so that an approximate optimal feedback can be computedat every point x, i.e. for the control problem we can define the feedback mapF : Ω → A. In order to construct F , we first introduce the notationI ∆ (x, a) := e −λ∆t I[V ](x + hf(x, a)) + ∆tg(x, a). (7.96)where the suffix ∆ indicates that this function also depends on the discretization.Note that I ∆ (x, ·) has a minimum over A, but the minimum point may not beunique.The possible occurrence of multiple minimizers requires to construct a selection,e.g. to take a strictly convex φ and defineThe selection is defined byA ∗ (x) = {a ∗ ∈ A : I ∆ x (x, a ∗ ) = minA I∆ (x, a)} (7.97)a ∗ x = arg min A ∗ (x) φ(a) (7.98)We are now able to compute our approximate optimal trajectories, by defining thepiecewise constant controla ∆ (s) = a ∗ x n,hs ∈ [t n , t n+1 ) (7.99)where y n; ∆t is the state of the Euler scheme, at the iteration n. Error estimatesof the approximation of feedbacks and optimal trajectories are available for controlproblems in [BCD97, F01].7.4.6 Numerical tests for control problemsWhile one-dimensional, academic examples are intended to complement the theoreticalpresentation of chapters 4 and 5, this chapter rather collects examples and testsrelated either to applications or to more challenging benchmarks. Among them, wemention the treatment of singular solutions in more than one space dimension, andDynamic Programming in dimension three and four.

✐✐7.4. SL schemes for optimal control problems 215W 1,∞ initial conditionn n L ∞ error L 1 error25 5.75 · 10 −2 6.57 · 10 −250 1.85 · 10 −2 2.76 · 10 −2100 6.17 · 10 −3 8.32 · 10 −3rate 1.61 1.49Table 7.1. Numerical errors for Test 1.Test 1: Quadratic Hamiltonian, Lipschitz continuous initial data This test refersto the HJ equation:{u t (x, t) + 1 2 |Du(x, t)|2 = 0 (x, t) ∈ [−2, 2] 2 × [0, 1]v(x, 0) = v 0 (x) = max(0, 1 − |x| 2 ) x ∈ [−2, 2] 2 (7.100)and is (along with the next example) a two-dimensional version of the test presentedin Chapter 5, which may be recast as a control problem via the definitions (7.53).The solution of this test problem generates a singularity in the gradient for t > 1/2.The exact solution of (7.100) can be explicitly computed and for t > 1/2 itreads{(|x|−1)2v(t, x) =2tif |x| ≤ 10 if |x| ≥ 1The solution, computed with ∆t = 0.1 and a cubic reconstruction, is shown in Figure7.1 at even time steps between t = 0 and t = 1. The graphs show in particularthat the singularity in the gradient is resolved isotropically and without undesirednumerical dispersion. Table 7.1 shows L ∞ and L 1 errors at T = 1 for differentnumbers of nodes n n on the edge of the computational domain. The solution iscomputed with good accuracy even with a low number of nodes, although therate of convergence is limited by the lack of semiconcavity, as it has already beenremarked in the one-dimensional examples of Chapter 5.Test 2: Quadratic Hamiltonian, semiconcave initial data We consider again theequation with quadratic Hamiltonian, but reversing the sign of the initial condition:{u t (x, t) + 1 2 |Du(x, t)|2 = 0 (x, t) ∈ [−2, 2] 2 × [0, 1]u(x, 0) = u 0 (x) = min(0, |x| 2 − 1) x ∈ [−2, 2] 2 (7.101).The exact solution of this test problem isu(t, x) ={|x|22t+1 − 1 if |x| ≤ √ 2t + 10 if |x| ≥ √ 2t + 1and generates no further singularity in the gradient. Figure 7.2 shows the numericalsolution computed for t ∈ [0, 1] with ∆t = 0.1 and a cubic reconstruction, and table7.2 shows L ∞ and L 1 errors at T = 1. In this case, due to the uniform semiconcavityof the solution, the improvement in L ∞ error is of one order of magnitude, and inL 1 error (which is clearly measured in a weaker norm) of more then two orders ofmagnitude.

✐✐216 Chapter 7. Control and Games110.50.500-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2(a)(b)110.50.500-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2(c)(d)110.50.500-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2(e)(f)Figure 7.1. Numerical solutions for Test 1Semiconcave initial conditionn n L ∞ error L 1 error25 1.57 · 10 −2 2.88 · 10 −250 4.03 · 10 −3 8.42 · 10 −4100 6.46 · 10 −4 3.29 · 10 −5rate 2.30 4.89Table 7.2. Numerical errors for Test 2.

✐✐7.4. SL schemes for optimal control problems 21700-0.5-0.5-1-1-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2(a)(b)00-0.5-0.5-1-1-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2(c)(d)00-0.5-0.5-1-1-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2-2-1.5-1-0.500.511.52 -2 -1.5-1 -0.50 0.5 1 1.5 2(e)(f)Figure 7.2. Numerical solutions for Test 2Test 3: the moon landing problem This is a classical control problem which istaken from [FR75]. A spacecraft is attempting to land smoothly on a surface usingthe minimum amount of fuel. The gravity acceleration g is assumed to be constantnear the surface. The motion is described by the equations⎧⎪⎨ ḣ = v˙v = −g + m⎪⎩−1 α(7.102)ṁ = −kαwhere h and v denote respectively the height and vertical velocity of the spacecraft,m is the mass of the spacecraft (including the variable mass of the fuel), and the

✐✐218 Chapter 7. Control and Gamescontrol α ∈ [0, ̂α] denotes the thrust of the spacecraft’s engine. The initial time ist 0 = 0 and the final time T is the first time the spacecraft touches the moon. Theinitial and final conditions for this problem readh(0) = h 0 , v(0) = v 0 , m(0) = M + F (7.103)h(T ) = 0, v(T ) = 0 (7.104)where h 0 and v 0 are the initial height and velocity of the spacecraft, M and Fare mass of the spacecraft without fuel and the initial mass of the fuel. Setting(x 1 , x 2 , x 3 ) = (h, v, m), the problem of minimizing the fuel is equivalent to minimize−m(T ) = −x 3 (T ) over the class of measurable controls which drive the system tothe point x 1 (T ) = x 2 (T ) = 0 and satisfy α(t) ∈ A = [0, ̂α].Treating the final condition by penalization to remove the target constraint,the control problem is in the Mayer formwhereJ x,t (α) = ψ(y x (T ; α)),ψ(x) = −x 3 + x2 1 + x 2 2(ɛ ≪ 1),ɛand its value function solves the Bellman equation⎧⎨−v t (x, t) + sup{−f(x, α) · Dv(x, t)} = 0α∈A⎩v(x, T ) = ψ(x).The test is performed using a P 2 reconstruction in the interpolation routine, and(following [FF94]) the Heun scheme to approximate the dynamics. In this versionof (7.77), the point x j + ∆tf(x j , a) is replaced by x j + ∆tΦ(∆t; x j , a 1 , a 2 ), whereΦ(∆t; x j , a 1 , a 2 ) = 1 2 [f(x j, a 1 ) + f(x j + ∆tf(x j , a 1 ), a 2 )] ,and the minimization is performed with respect to both a 1 and a 2 . For each pointof the space-time grid the approximate optimal feedback is defined by taking thearg min a1,a 2in (7.77) and extending the two discrete controls to all the computationaldomain by interpolation.The test uses the initial condition (x 1 , x 2 , x 3 ) = (0.3, 0, 18), final time T = 1,space steps of (0.03, 0.15, 0.26) and a time step of ∆t = 0.03. Figure 7.3 shows theoptimal trajectory in the phase space (velocity, position), along with the discreteoptimal controls a 1 and a 2 during the time interval [0, 1]. The spacecraft actuallylands at t = 0.99, and the end point (x 1 , x 2 ) = (0, 0) is reached with some numericalerror, due to the penalization strategy. All figures show that the optimal trajectoryis basically obtained falling down by gravity up to a switching time t ∗ , and thenbraking until the landing with maximum thrust. This is in perfect agreement withtheory.Test 4: a Ramsey model for a multi-sectors economy Consider an economywhich has three sectors S i , i = 1, . . . , 3 (the model can be easily extended to manysectors) and denote by Y i (s), K i (s), L i (s) the output rate, the capital and the labourof sector S i at time s. Capital and labour are connected to the production through

✐✐7.4. SL schemes for optimal control problems 21930"controls1"30"controls2"25252020controls15controls15101055000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1time(a)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(b)time0.3’trajectory’0.250.20.15height0.10.050-0.05-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1velocity(c)Figure 7.3. Moon Landing, (a): optimal control a 1 , (b): optimal controla 2 , (c): optimal trajectory.a production function Y i = F i (K i , L i ). A classical example is the Cobb–DouglasfunctionF (K, L) = e λs K α L 1−α , (0 ≤ α ≤ 1) (7.105)where λ is a coefficient connected to the technical innovation (see [1]). For everysector, we can divide by L i and define y i = YiL i, k i = KiL i, i = 1, . . . , 3 obtaining thepro-capite production functionsy i = e λis k αii , (0 ≤ α i ≤ 1) (7.106)A typical choice is for the parameters λ i is λ i = m(1 − α i ) (see [1]), where m is aconstant rate of technical innovation.The rate of consumption c(s) is a fraction u (the control) of a function g(k(s))of the capital, i.e., for every sector the consumption rate is given byc i (s) = u(s)g i (k i (s)), (u min ≤ u(s) ≤ 1). (7.107)Then, the rate of change of the capital in every sector (assuming that the sectorsare separated) is˙k i = f(k i ) − ug(k i ), (u min ≤ u ≤ 1). (7.108)

✐✐220 Chapter 7. Control and Games’controls’10.80.60.40.200 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Figure 7.4. Ramsey model, optimal control.Let now U be a utility function which represents the utility of consuming at ratec = c 1 + c 2 + c 3 > 0. This function U is typically monotone increasing and boundedon the positive axis. The goal is to maximize the functional∫ T0U(c(s))ds, (7.109)with respect to the choice of the control u under the constraints k i (0) = k 0 i , k i(T ) =k 1 i .Figures 7.4 and 7.5 show respectively the optimal control u and the evolutionin time of the variables k i and c i (i = 1, 2, 3). The various parameters have beenset as α 1 = 0.2, α 2 = 0.5 and α 3 = 0.8, u min = 0.1, m = 1, whereas g is defined asand the utility function U asg(k) = k − k min , (7.110)U(c) = Mc21 + c 2 , (7.111)with k min = k 0 /3 and M = 10. The initial condition for the three components ofthe capital is k 0 = (6, 4, 2) and the final objective for T = 1.8 is k 1 = (8, 6, 4). Asin the previous test, we treat the final constraint by penalization, using the finalcondition∣ ∣ k − k1 2ψ(k) = −ɛFor the computation, we have used ∆t = 0.05, ɛ = 10 −4 and 30 nodes in space forevery dimension.Figures 7.4 and 7.5 depict a situation in which the capital increase, with consumptionat its minimum, up to a time t ≈ 1.2. After t, the control switches from0.1 to 1, the consumption go to its maximum and the first two component of the capitalstart decreasing. The final position of the trajectory is k(T ) ≈ (7.91, 7.41, 5.68),and significantly differs from the target, due to both penalization and lack of controllability(a one-dimensional control has been used for a system in R 3 ).Test 5: control of the HIV-1 dynamics In the paper [PN99] Perelson and Nelsonhave described the dynamics of the HIV-1 virus infection (the virus that causes

✐✐7.4. SL schemes for optimal control problems 221’trajectory_k_1’7’consumption_c_1’8.56857.543726.5160 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8(a)00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8(d)8’trajectory_k_2’7’consumption_c_2’7.56756.54635.5524.5140 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8(b)00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8(e)6’trajectory_k_3’5.5’consumption_c_3’5.554.5544.53.5342.53.5231.512.50.520 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8(c)00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8(f)Figure 7.5. Ramsey model, optimal solution, (a)–(c): k i , (d)–(f): c i .AIDS). In this test, we add a control term in their model in order to describe thedrug therapy effect.The model considers three populations: the uninfected cells C, the productivelyinfected cells I and the virus V . All the populations are described by their density

✐✐222 Chapter 7. Control and Gamesper volume unit. The dynamics is described by the system⎧(⎪⎨ Ċ = s + pC 1 − C )− d C C − (1 − u)kV C,C maxI ˙ = (1 − u)kV C − d I I,⎪⎩˙V = Nd I I − d V V,(7.112)where s represents the rate at which new C cells are created from sources withinthe body, p is the maximum proliferation rate for the C cells, C max is the maximumcell population density, d C , d I and d V are the death rates for respectively the C,I and V populations, Nd I is the average rate of virion production, u is the controldriven by the drug therapy (usually, the constraint d C C max > s is added). Theterm kV C tells that the rate of creation of new infected cell is proportional to theproduct V C (at least one virion should enter a cell to infect it).A typical drug therapy is a treatment based on RT (reverse transcriptase) or proteaseinhibitors. The control u measures the effectiveness of the inhibitor: for u = 1the inhibition is 100% effective, whereas for u = 0 there is no inhibition. Typically,0 ≤ u ≤ u max < 1. The functional we want to minimize has the form∫ T0(M(I 2 + V 2) + u ) dt (7.113)(for a positive constant M) and takes into account the dimension of the populationsI and V and the fact that a drug excess can have negative global effects on thepatient.In the test, parameters have been set as s = 0.2, p = 0.05, k = 0.08, d C = 0.01,d I = 0.3, d V = 0.009, N = 0.005 and M = 8 (a biologically realistic choice ofsome of the parameters can be found in [2]). Figure 7.6 shows the optimal controlu and the time evolution of the state variables C, I, V . The virus populationdecreases monotonically under the effect of the drug therapy, but since the use oflarge amounts of drug is penalized, the optimal therapy suggested by the model isto use the drug at its maximum level for the first period and then stop the druginjection. After this first time interval, the evolution of infected cells and virusesnaturally decreases by the uncontrolled dynamics.Tests 6–7: the minimum time problem In the first test, the system has a variablevelocity defined byf(x, a) = c(x) a = |x 1 + x 2 | a,with a ∈ B(0, 1). It has to be driven to the origin, and more precisely to the targetT = B(0, ε),where, in the numerical computation, we have set ε = ∆x/2.Figure 7.7 shows the approximate minimum time function T (x) and its levelscurves, whereas the optimal trajectories (not shown in the figure) are orthogonal tothe level curves. On the line x = −y the velocity drops to zero and we would haveT = +∞. Note that for points close to this line, the optimal trajectory first movesto a region in which the velocity is higher, then turns towards the target.

✐✐7.4. SL schemes for optimal control problems 223’controls’9.25’trajectory_C’19.20.80.69.150.49.10.29.0500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(a)90 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(b)0.82’trajectory_I’0.202’trajectory_V’0.80.20.780.1980.760.1960.740.1940.720.1920.70.190.680.1880.660.1860.640 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(c)0.1840 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(d)Figure 7.6. HIV-1 dynamics, (a): optimal control, (b): uninfected cells,(c): infected cells, (d): virus.281.571650.5403220.510210122011.522 1.5 1 0.5 0 0.5 1 1.5 2Figure 7.7. Minimum time function T , graph (left) and contours (right).

✐✐224 Chapter 7. Control and Games1.51.5110.50.5000.50.511.51.5 1 0.5 0 0.5 1 1.5griglia= 100x100110griglia= 100x10011.5 1 0.5 0 0.5 1 1.5Figure 7.8. Level sets of T (left) and an optimal trajectory on the surfacez (right).The second test for the minimum time problem is related to the computationof geodesics on a nonsmooth surface. In this test, we set a point target:on the pyramidal surfacez(x) =T = (0, −0.6, 0.4){1 − (|x 1 | + |x 2 |) if |x 1 | + |x 2 | < 10 elsewhere.The system moves on the surface with intrinsic velocity in B(0, 1), so that theminimum time function (restricted to the surface) is in fact the intrinsic distance,and optimal trajectories are geodesics of the surface. In this test, we compute thegeodesic reaching the target from the starting point (0, 0.5, 0.5).It can be shown that the 3D problem can be reduced to a 2D problem bymodifying the velocity field according to the function z. In fact, in case of a unityintrinsic velocity on the surface, it can be shown (see [Se96, Se96b]) that the velocityof the corresponding 2D problem becomesc(x, a) =1√1 + (Dz · a)2 .Figure 7.8 shows the level sets of T and its surface with the optimal trajectory on it.7.5 Problems with state constraintsMany relevant applications in control theory require the state of the system to satisfyadditional constraints – typically, to remain within some prescribed set of the statespace R d . We turn now our attention to this case, trying to give a presentation ofthe general ideas while skipping the most delicate technical points. The interestedreader will find suitable references for state constrained problems in the referencesection at the end of this chapter.

✐✐7.5. Problems with state constraints 225Consider our standard dynamics (7.2) coupled with the infinite horizon costfunctional (7.5). In their most typical form, the state constraints require that, forall s > 0,y x (s; α) ∈ Ω, (7.114)where we assume thatΩ is a bounded, open subset of R d . (7.115)While, in the case of free control problems, the cost J x is minimized over the setA of measurable controls taking values in A, condition (7.114) restricts admissiblecontrols to a new set which will depend on both the constraint Ω and the initialposition x of the system:A x = {α ∈ A : y x (s; α) ∈ Ω for all s > 0}. (7.116)This set will be assumed to be nonempty for all x ∈ Ω.Moreover, in general the “state constrained” value functionv(x) = infα∈A x∫ ∞0g(y x (s; α))e −λs ds (7.117)can happen to be discontinuous, due to the complex relationship between the dataof the problem (dynamics and constraints) and the set A x . Let us denote by A x ,for x ∈ ∂Ω, the subset of controls such that the corresponding vectorfield f pointsinside the constraint, i.e.{Aif x ∈ ΩA x =(7.118){a ∈ A : f(x, a) · ν(x) < 0} if x ∈ ∂Ω.Clearly, this set coincides with the usual control set A at interior points, where theconstraint is not active. Note that the map x → A x is not regular in general, as thefollowing simple example shows.Consider the dynamics in R 2 :ẏ(s) = α(s) ∈ A = B(0, 1),and Ω = (−1, 1) 2 . The system can move in every direction at internal points,whereas on the right hand side of the square the admissible controls can only bechosen in the half ball B(0, 1)∩{a 1 < 0}, on the upper side of the square in the halfball B(0, 1) ∩ {a 2 < 0} and so on. At the vertices of Ω, admissible controls must bechosen in a quadrant of B(0, 1).It has been shown by Soner [So86a, So86b] that the value function is continuous (andtherefore, uniformly continuous) on Ω if for some positive constant β the followingboundary condition on the vector field f is satisfied:For all x ∈ ∂Ω there exists a ∈ A such that f(x, a) · ν(x) ≤ −β < 0, (7.119)where ν(x) is the outward normal to Ω at the point x. Condition (7.119) is satisfiedin the example above. However, if the state constraint Ω is cut along the x 2 axisto obtain the new constraint ˜Ω = Ω \ {(0, x 2 ) : −1 ≤ x 2 ≤ 0} and the dynamics isdefined by ẏ = (a 1 , 0) with a 1 > 0, it is easy to see that starting at an initial pointon the left of the cut the dynamics will remain trapped on the left, and even for a

✐✐226 Chapter 7. Control and Gameslinear running cost such as g(x 1 , x 2 ) = 1 − x 1 we will have a discontinuity of thevalue function inside ˜Ω. Note that condition (7.119) is not satisfied in this latterexample, although the set A x is always nonempty.Under the condition (7.119), by the Dynamic Programming Principle, Sonerhas shown that the value function v is the unique “state constrained” viscositysolution ofH(x, u(x), Du(x)) = 0, x ∈ Ω, (7.120)which in turn means that v satisfies (in the viscosity sense) the inequalitiesH(x, u(x), Du(x)) ≤ 0 x ∈ Ω (7.121)H(x, u(x), Du(x)) ≥ 0 x ∈ Ω (7.122)with H defined by (7.18). In other terms, the value function v is a solution insideΩ, but only a supersolution on the boundary ∂Ω.Theorem 7.17. Let the standard assumptions on f and g be satisfied, and letv ∈ C(Ω). Then, v is the unique “state constrained” viscosity solution of (7.120)on Ω.Rather than giving the proof of this theorem, we show the underlying heuristicargument. In our previous examples we have seen that at all internal point we haveA x = A since the dynamics can move without restrictions. This implies that vshould satisfy the same equation as in the unconstrained case. On the other hand,if x ∈ ∂Ω, then the constraint reduces the set of admissible directions, so thatA x ⊂ A and we haveH(x, v(x), Dv(x)) = λv(x) + max{−f(x, a) · Dv(x) − g(x, a)} ≥a∈A≥ λv(x) + max{−f(x, a) · Dv(x) − g(x, a)} = 0. (7.123)a∈A xThe value function should therefore be a supersolution for (7.120).Remark 7.18 (Necessary and sufficient conditions). Condition (7.119) isknown to be only a sufficient condition for the existence of trajectories which remainwithin Ω. However, necessary and sufficient condition for the existence of solutionsin Ω have been extensively studied in viability theory (see [A91].In order to simplify the presentation, let Ω be an open convex subset of R d . Atrajectory is called viable wheny(s) ∈ Ω, ∀s ≥ 0. (7.124)Let F : Ω → R d be a multivalued map, assumed to be lower semicontinuous and withcompact convex images (we refer to [AC84] for the theory and definitions related tomultivalued maps). Define the tangent cone to Ω at the point x, as( )⋃T K (x) := µ(K − x) . (7.125)µ>0A result due to Haddad [Ha81] shows that the conditionF (x) ∩ T K (x) ≠ ∅, ∀x ∈ Ω, (7.126)

✐✐7.5. Problems with state constraints 227is necessary and sufficient to have viable trajectories for the multivalued Cauchyproblem{ẏ(s) ∈ F (y(s)) s ≥ 0(7.127)y(0) = x.This result has also been extended to more general sets and more general tangentcones.Time and space discretization In order to build a discretization of (7.120) we usethe standard discretization in time (7.54), (7.56) for respectively the dynamics andthe cost functional. For x ∈ Ω, the corresponding value function for the discretetime problem isv ∆t (x) = inf J ∆tα ∆t x (α ∆t ), (7.128)where α ∆t ∈ A ∆tx , the set of discrete control sequences {a m } such thata m ∈ A ∆tx = {a ∈ A : x + ∆tf(x, a) ∈ Ω}. (7.129)By standard arguments, we can obtain also for the state constrained problem aDynamic Programming Principle, in the form{}∑¯n−1v ∆t (x) = inf ∆t g(y m , a m )e −λtm + e −λt¯n v ∆t (y¯n ) . (7.130)α ∆t m=0for all x ∈ R d and any positive integer ¯n. Setting as usual ¯n = 1, this gives thetime-discrete schemev ∆t (x) =inf{∆tg(x, a) + e −λ∆t v ∆t (x + ∆tf(x, a)) } (7.131)a∈A ∆t xand finally, replacing the computation of v ∆t (x + ∆tf(x, a)) with an interpolation,the fully discrete schemev j = mina∈A ∆t x j{∆tg(xj , a) + e −λ∆t I 1 [V ](x j + ∆tf(x j , a)) } (7.132)written at the grid node x j . An iterative version can also be derived, as in theunconstrained case.Convergence analysis It can be proved that for v ∈ C 0,γ (Ω) and bounded variationoptimal controls the following estimate holds true‖v − v ∆t ‖ ∞ ≤ C∆t γ (7.133)Finally, for the fully discrete scheme the following a-priori estimate holds true‖V ∆t − V ‖ ∞ ≤where ω(∆t) denotes the modulus of continuity of v ∆t .ω(∆x) . (7.134)1 − e−λ∆t

✐✐228 Chapter 7. Control and Games7.6 Dynamic Programming for differential gamesThe Dynamic Programming approach can be also be applied applied to the analysisand approximation of differential games. . In the general setting, we consider thenonlinear system { ẏ(t) = f(y(t), a(t), b(t)), t > 0,(7.135)y(0) = xwhere y(t) ∈ R d is the statea( · ) ∈ A is the control of player 1 (player a)b( · ) ∈ B is the control of player 2 (player b),A{ a : [0, +∞[ → A, measurable } (7.136)B = { b : [0, +∞[ → B, measurable }, (7.137)A, B ⊂ R M are given compact sets. A typical choice is to take as admissible controlfunction for the two players piecewise constant functions respectively with valuesin A or B. Assume f is continuous and|f(x, a, b) − f(y, a, b)| ≤ L |x − y| ∀x, y ∈ R N , a ∈ A, b ∈ B.By Caratheodory’s theorem the choice of measurable controls guarantees that forany given a( · ) ∈ A and b( · ) ∈ B, there is a unique trajectory of (7.135) which wewill denote by y x (t; a, b). The payoff can be defined as in control problems. Forexample, for the infinite horizon problem we will haveJ x (a(·), b(·)) :=∫ +∞0g(y x (s; a, b))e −λs ds (7.138)The typical situation described by differential games is when player-a wants tominimize the payoff and player-b wants to maximize the payoff. So it is natural todefine the value of the game asv(x) := infsupa(·) b(·))J x (a(·), b(·)). (7.139)This is the case for 0 − sum differential games, where the gain of one player correspondsto the loss of the other player. This class of games is simpler with respect tothe general class of noncooperative N-players games where Nash theory of equilibriacan be applied.However, note that due to the presence of two players it is delicate to definewho is choosing first since, in general, we know thatinf supa(·) b(·)J x (a(·), b(·)) does not coincide with sup inf J x(a(·), b(·)) (7.140)b(·) a(·)We will present the main ideas of this approach in the simplified case of pursuitevasiongames .Example 7.19 (Pursuit-Evasion Games) These are particular differential gameswhere, given a closed target T ⊆ R N we define the payoff ast x (a( · ), b( · )) = min{ t : y x (t; a, b) ∈ T } ≤ +∞, (7.141)

✐✐7.6. Dynamic Programming for differential games 229xx yy EE PPFigure 7.9. Zermelo navigation problemNaturally, t x {a(·), b(·)} will be finite only under additional assumptions on thetarget and on the dynamics. The two players are opponents since player a wants tominimize the payoff (he is called the pursuer) whereas player b wants to maximizethe payoff (he is called the evader). The dynamics of these games can be splittedsince each each player is just controlling its own dynamics. So we assume that thedimension d of the state space is even and write the dynamics as{ẏ1 = f 1 (y 1 , a), y i ∈ R d/2 , i = 1, 2(7.142)ẏ 2 = f 2 (y 2 , b)For this game, the target will be given byT ɛ ≡ { |y 1 − y 2 | ≤ ɛ }, for ɛ > 0, or T 0 ≡ { (y 1 , y 2 ) : y 1 = y 2 } .Then, t x (a( · ), b( · )) is the capture time corresponding to the strategies a(·) andb(·).Example 7.20 (Zermelo navigation problem) A boat B moves with constantvelocity in a river and it can change its direction istantaneously. The water of theriver flows with a velocity σ and the boat tries to reach an island in the middle ofthe river (the target) maneuvering against water current. We choose a system ofcoordinates such that the velocity of the current is (σ, 0) (see Figure 7.9). In thenew system, the dynamics of the boat is described by{ẋ = σ + v B cos a,ẏ = v B sin a,where a ∈ [−π, π] is the control over the boat direction. Let z(t) = ( x(t), y(t) ) . Itis easy to see that v B > σ is a sufficient condition for the boat to reach the islandfrom any initial condition in the river. This is not the case if the opposite conditionholds true (see below). Let us introduce the reachable setR ≡ { x 0 : ∃t > 0, a(·) ∈ A such that y(t; x 0 , a(·)) ∈ T } . (7.143)This is the set of initial positions from which the boat can reach the island. Let uschoose T = { (0, 0) } , then a simple geometric argument shows that⎧⎪⎨ R 2 se v B > σ,{ }R = (x, y) : x < 0 or x = y = 0 if v B = σ,⎪⎩ { }(x, y) : x < 0, |y| ≤ −x vB (σ 2 − vB 2 )− 1 2 if 0 ≤ vB < σ.(7.144)The result is obvious for v B ≥ σ. Now assume 0 < v B < σ. The motion of theboat at every point is determined by the (vector) sum of its velocity v B and thecurrent velocity σ. (Figure 7.10). The maximum angle which is allowed to the boat

✐✐230 Chapter 7. Control and Gamesdirection is≡ π 2 + arctan (v B√σ2 − v 2 B)and the equation of the line with this slope passing from the origin isy = −xv B√σ2 − v 2 B,which explains (7.144).7.6.1 Dynamic Programming for gamesThe first question is: how can we define the value function for the 2-players game? Certainly it is notinf sup J x (a, b)a∈A b∈Bbecause a would choose his control function with the information of the wholefuture response of player b to any control function a( · ) and this will give him a bigadvantage. A more unbiased information pattern can be modeled by means of thenotion of nonanticipating strategies (see [EK72] and the references therein),∆ ≡ { α : B → A : b(t) = ˜b(t) ∀t ≤ t ′ ⇒ α[b](t) = α[˜b](t) ∀t ≤ t ′ }, (7.145)Γ ≡ { β : A → B : a(t) = ã(t) ∀t ≤ t ′ ⇒ β[a](t) = β[ã](t) ∀t ≤ t ′ } . (7.146)The above definition is fair with respect to the two players. In fact, if player achooses his control in ∆ he will not be influenced by the future choices of player b(Γ has the same role for player b). Now we can define the lower value of the gameorT (x) ≡ infv(x) ≡ infsupα∈∆ b∈Bsupα∈∆ b∈Bt x (α[b], b),J x (α[b], b)when the payoff is J x (a, b) = ∫ t x(a,b)e −t dt. Similarly the upper value of the game0is˜T (x) ≡ sup inf t x(a, β[a]),a∈Aorβ∈Γṽ(x) ≡ sup inf J x(a, β[a]) .a∈Aβ∈ΓWe say that the game has a value if the upper and lower values coincide, i.e. ifT = ˜T or v = ṽ.Lemma 7.21 (DPP for games). For all 0 ≤ t < T (x)T (x) = infsupα∈∆ b∈B{ t + T (y x (t; α[b], b)) }, ∀x ∈ R \ T ,andv(x) = infα∈∆ supb∈B{ ∫ t0}e −s ds + e −t v(y x (t; α[b], b)) , ∀x ∈ T c .

✐✐7.6. Dynamic Programming for differential games 231The proof is similar to the 1-player case but more technical due to the use ofnon-anticipating strategies . Note that the upper values ˜T and Ṽ satisfy a similarDPP. Let us introduce the two Hamiltonians for games Isaacs’ Lower HamiltonianIsaacs’ Upper HamiltonianH(x, p) ≡ minb∈B˜H(x, p) ≡ maxa∈AThe following theorem can be found in [ES84].max { −p · f(x, a, b) } − 1 .a∈Amin { −p · f(x, a, b) } − 1 .b∈BTheorem 7.22. 1. If R\T is open and T ( · ) is continuous, then T ( · ) is a viscositysolution ofH(x, ∇T ) = 0 in R \ T . (7.147)2. If v( · ) is continuous, then it is a viscosity solution ofv + H(x, ∇v) = 0 in T c .As we already said, the main issue in this theory of weak solutions is to proveuniqueness results. This point is very important also for numerical purposes sincethe fact that we have a unique solution allow to prove convergence results for theapproximation schemes. To this end, let us consider the Dirichlet boundary valueproblem { u + H(x, ∇u) = 0 in Ω(7.148)u = gon ∂Ωand prove a uniqueness result under assumptions on H including Bellman’s HamiltonianH 1 . This new boundary value problem is connected to (7.37) because thenew solution of (7.148) is a rescaling of T . In fact, introducing the new variablev(x) ≡{1 − e−T (x)if T (x) < +∞, i.e. x ∈ R1 if T (x) = +∞, (x /∈ R)(7.149)it is easy to check that, by the DPP,v(x) =inf J x(a)a( · )∈AwhereMoreover, V is a solution ofJ x (a) ≡∫ tx(a)0e −t dt .{v(x) + max { −∇v(x) · f(x, a) − 1 } = 0 ina∈Av(x) = 0RN \ Tx ∈ ∂T(7.150)which is a special case of (7.148), with H(x, p) = H 1 (x, p) ≡ max a∈A { −p · f(x, a) −1 } and Ω = T c ≡ R N \ T . The change of variable (7.41) is called Kružkov transformationand has several advantages. First of all V takes values in [0, 1] whereas T

✐✐232 Chapter 7. Control and Gamesis generally unbounded and this helps in the numerical approximation. Moreover,one can always reconstruct T and R from V by the relationsT (x) = − log(1 − v(x)), R = { x : v(x) < 1 }.Lemma 7.23. The Mininimum Time Hamiltonian H 1 satisfies the “structuralcondition”|H(x, p) − H(y, q)| ≤ K(1 + |x|)|p − q| + |q| L |x − y| ∀x, y, p, q, (7.151)where K and L are two positive constants.The following theorem has been proved in [CL83].Theorem 7.24. Assume H satisfies (SH), u, w ∈ BUC(¯Ω), u subsolution, wsupersolution of v + H(x, ∇v) = 0 in Ω (open), u ≤ w on ∂Ω. Then, u ≤ w in Ω.Definition 7.25. We call subsolution (respectively supersolution) of (7.42) a subsolution(respectively supersolution) u of the differential equation such that u ≤ 0on ∂Ω (respectively ≥ 0 on ∂Ω).Corollary 7.26. If the value function V ( · ) ∈ BUC(T c ), then V is the maximalsubsolution and the minimal supersolution of (7.42) (we say it is the completesolution). Thus V is the unique viscosity solution.If the system is STLC around T (i.e. T ( · ) is continuous at each point of ∂T )then V ∈ BUC(T c ) and we can apply the Corollary.The structural condition (7.151) plays an important role for uniqueness.Lemma 7.27. Isaacs’ Hamiltonians H, ˜H satisfy the structural condition (7.151)Then Comparison Theorem 7.24 applies and we getTheorem 7.28. If the lower value function v( · ) ∈ BUC(T c ), then v is the completesolution (maximal subsolution and minimal supersolution) of{ u + H(x, Du) = 0 in T c ,. (7.152)u = 0 on ∂T .Thus V is the unique viscosity solution.Note that for the upper value functions ˜T and ˜W the same results are validwith H = ˜H. We can give “capturability” conditions on the system ensuringv, ṽ ∈ BUC(T c ). However, those conditions are less studied for games becausethere are important pursuit-evasion games with discontinuous value, the games with“barriers” (see the monography [I65] for an extensive presentation of differentialgames). It is important to note that in general the upper and the lower values aredifferent. However, the Isaacs conditionH(x, p) = ˜H(x, p) ∀x, p, (7.153)

✐✐7.6. Dynamic Programming for differential games 233guarantees that they coincide.Corollary 7.29. If v, ṽ ∈ BUC(T c ), thenv ≤ ṽ, T ≤ ˜T .If Isaacs condition holds then v = ṽ and T = ˜T , (i.e. the game has a value).Proof. Immediate from the comparison and uniqueness for (7.148).For numerical purposes, one can decide to write down an approximationscheme for either the upper or the lower value using the techniques of the nextsection. Before going to it, let us give some informations about the characterizationof discontinuous value functions for games. This is an important issue becausediscontinuities appear even in classical pursuit-evasion games (e.g. in the homicidalchauffeur game that we will present later). We will denote by B(Ω) the set ofbounded real functions defined on Ω. Let us start with a definition which has beensuccessfully applied to convex Hamiltonians. As we have seen, to obtain uniquenessone should prove that a comparison principle holds, i.e. for every subsolution wand supersolution W we havew ≤ WAlthough this is sufficient to get uniqueness in the convex case the above definitionwill not guarantee uniqueness for nonconvex hamiltonians (e.g. min-max Hamiltonians).Two new definitions have been proposed. Let us denote by S the set of subsolutionsof our equation and by Z the set of supersolutions always satisfying the Dirichletboundary condition on ∂Ω.Definition 7.30 (minmax solutions [Su95]). u is a minmax solution if thereexists two sequences w n ∈ S and W n ∈ Z such thatw n = W n = 0 on ∂Ω (7.154)w n is continuous on ∂Ω (7.155)and limnw n (x) = u(x) = limnW n (x), x ∈ Ω. (7.156)Definition 7.31 (e-solutions, see e.g. [BCD97]). u is an e-solution (envelopesolution) if there exists two non empty subsetsS(u) ⊂ SZ(u) ⊂ Zsuch that ∀ x ∈ Ωu(x) =sup w(x) = inf W (x)w∈S(u)W ∈Z(u)By the comparison Lemma there exists a unique e-solution.In fact, if u and v are two e-solutions,u(x) =sup w(x) ≤ inf W (x) = v(x)w∈S(u)W ∈Z(v)

✐✐234 Chapter 7. Control and Gamesand alsov(x) =sup w(x) ≤ inf W (x) = u(x)w∈S(v)W ∈Z(u)It is interesting to note that in our problem the two definitions coincide.Theorem 7.32. Under our hypotheses, u is a minmax solution if and only if u isan e-solution.Time discretization for games The same time discretization can be written forthe dynamics and natural extensions of N and v ∆t are easily obtained. The crucialpoint is to prove that the discrete DPP holds true and that the upper value of thediscrete game is the unique solution in L ∞ (R d ) of the external Dirichlet problemv ∆t (x) = S(v ∆t )(x) on R d \ T (7.157)v ∆t (x) = 0 on ∂T (7.158)where the fixed point operator now isS(v ∆t )(x) ≡ max min[e −∆t v ∆t (x + hf(x, a, b)) ] + 1 − e −∆tb∈B a∈AThe next step is to show that the discretization (7.157)–(7.158) is convergent tothe upper value of the game. A detailed presentation goes beyond the purposes ofthis paper, the interested reader will find these results in [BS91b]. We just give themain convergence result for the continuous case.Theorem 7.33. Let v ∆t be the solution of (7.157)–(7.158), Let T be compact withnonempty interior, the assumptions on f be verified, v be continuous.Then, v ∆tconverges to v locally uniformly in R d for ∆t → 0 + .Space discretization for differential games Let us get back to zero-sum games.Using the change of variable v(x) ≡ 1 − e −T (x) we can set the Isaacs equation in R dobtaining{v(x) + minv(x) = 0b∈B maxa∈A[−f(x, a, b) · ∇v(x)] = 1 inRd \ Tfor x ∈ ∂T(7.159)Assume we want to solve the equation in Q, an hypercube in R N . As we have seen,in the case of a single player we need to impose boundary conditions on ∂Q or, atleast, on I out . However, the situation for games is much more complicated. In fact,setting the value of the solution outside Q equal to 1 (as in the single player case)will imply that the pursuer looses every time the evader drives the dynamics outsideQ. On the contrary, setting the value to 0 outside Q will give a great advantage tothe pursuer. One way to define more unbiased boundary conditions is the following.Assume that Q = Q 1 ∩ Q 2 , where Q i , i = 1, 2 are subsets of R N/2 which can beinterpreted as the set of constraints for the i-th player. For example, in R 2 we canconsider as Q 1 a vertical strip and as Q 2 an horizontal strip and compute in therectangle Q which is the intersection of those strips. According to this construction,we penalize the pursuer if the dynamics exits Q 1 and the evader if the dynamicsexits Q 2 . When the dynamics exits Q 1 and Q 2 we have assign a value, e.g. giving

✐✐7.6. Dynamic Programming for differential games 235an advantage to one of them (in the following scheme we are giving advantage tothe evader). The discretization in time and space leads to a fully discrete schemewhere β ≡ e −∆t andw(x i ) = max min[βw(x i + hf(x i , a, b))] + 1 − β for i ∈ I in (7.160)b aw(x i ) = 1 for i ∈ I out2 (7.161)w(x i ) = 0 for i ∈ I T ∪ I out1 (7.162)I in = {i : x i + hf(x i , a, b) ∈ Q \ T for any a ∈ A, b ∈ B} (7.163)I T = {i : x i ∈ T ∩ Q} (7.164)I out1 = {i : x i /∈ Q 2 } (7.165)I out2 = {i : x i /∈ Q 2 \ Q} (7.166)Theorem 7.34. The operator S defined in (7.160) has the following properties:i) S is monotone, i.e. U ≤ V implies S(U) ≤ S(V );ii) S : [0, 1] L → [0, 1] L ;iii) S is a contraction mapping in the max norm,‖S(U) − S(V )‖ ∞ ≤ β‖U − V ‖ ∞The proof of the above theorem is a generalization of that related to theminimum time problem and can be found in [BFS94]. The above result guaranteesthat there is a unique fixed point U ∗ for S. Naturally the numerical solution w willbe obtained extending by linear interpolation the values of U ∗ and it will dependon the discretization steps ∆t and ∆x. Let us state the first convergence result forcontinuous value functionsTheorem 7.35. Let T be the closure of an open set with Lipschitz boundary, “diamQ → +∞” and v be continuous. Then, for ∆t → 0 + and ∆x∆t → 0+ , w ∆ convergesto v locally uniformly on the compact sets of R d .Note that the requirement “diam Q → +∞” is just a technical trick to avoidto deal with boundary conditions on Q (a similar statement can be written in thewhole space just working on an infinite mesh). We conclude this section quoting aconvergence result which holds also in presence of discontinuities (barriers) for thevalues function. Let w ɛ n be the sequence generated by the numerical scheme withtarget T ɛ = {x : d(x, T ) ≤ ɛ}.Theorem 7.36. For all x there exists the limitw(x) =lim wn(x)ɛε→0 +n→+∞n≥n(ɛ)and it coincides with the lower value V of the game with target T , i.e. w = V .Convergence is uniform on every compact set where V is continuous.

✐✐236 Chapter 7. Control and GamesCan we know a-priori what is the accuracy of the method in the approximationof the value function? This result is necessary to understand how far we are fromthe real solution when we compute our approximate solution. To simplify, let usassume that the Lipschitz constant for f L f ≤ 1 and that v is Lipschitz continuous.Then,( ) ) 2 ∆x‖w h,k − v‖ ∞ ≤ C∆t(1 1/2 +∆tThe proof of the above error estimate is rather technical and can be found in [Sor98].Approximation of optimal feedback and trajectories for games The numericalsolution w ∆ has been extended to Q by interpolation we can also compute anapproximate optimal feedback at every point of x ∈ Q, i.e. for the control problemwe can define the feedback map F : Q → A. For games, the algorithm computes anapproximate optimal control couple (a ∗ , b ∗ ) at every point of the grid. Again by wwe can also compute an approximate optimal feedback at every point x ∈ Q.(a ∗ (x), b ∗ (x)) ≡ argminmax{e −∆t w(x + hf(x, a, b))} + 1 − e −∆t (7.167)If that control is not unique then we can select a unique couple, e.g. minimizing twoconvex functionals. A typical choice is to introduce an inertial criterium to stabilizethe trajectories, i.e. if at step n + 1 the set of optimal couples contains (a ∗ n, b ∗ n) wekeep it.7.6.2 Numerical tests for pursuit-evasion gamesLet us examine some classical games and look at their numerical solutions. We willfocus our attention to the accuracy in the approximation of the value function aswell as to the accuracy in the approximation of optimal feedbacks and trajectories.In the previous sections we always assumed that the sets of controls A and B werecompact. In the algorithm and in the numerical tests we have used a discrete finiteapproximation for those sets which allows to compute the min-max by comparison.For example, we will consider the following discrete sets{A = a 1 + j a 2 − a}1, j = 0, . . . , c − 1;{B =c − 1b 1 + j b 2 − b 1c − 1}, j = 0, . . . , c − 1;where [a 1 , a 2 ] and [b 1 , b 2 ] represent the control sets respectively for the pursuer Pand the evader E. Finally, note that all the value functions represented in thepictures have values in [0, 1] because we have computed the fixed point after theKružkov change of variable.7.6.3 The Tag-Chase GameTwo boys P and E are running one after the other in the plane R 2 . P wants tocatch E in minimal time whereas E wants to avoid the capture. Both of them arerunning with constant velocity and can change their direction instantaneously. Thismeans that the dynamics of the system isf P (y, a, b) = v P af E (y, a, b) = v E b

✐✐7.6. Dynamic Programming for differential games 237where v P and v E are two scalars representing the maximum speed for P and E andthe admissible controls are taken in the setsA = B = B(0, 1).Let us give a more explicit version of the dynamics which is useful for the discretization.Let us denote by (x P , y P ) the position of P and by (x E , y E ) the position ofE, we can write the dynamics as⎧⎪⎨⎪⎩ẋ P = v P sin θ Pẏ P = v P cos θ Pẋ E = v E sin θ Eẏ E = v E cos θ E(7.168)where θ P ∈ [a 1 , a 2 ] ⊆ [−π, π] is the control for P and θ E ∈ [b 1 , b 2 ] ⊆ [−π, π] is thecontrol for E, θ P and θ E are the angles between the y axis and the velocities for Pand E (see Figure 7.6.3).We say that E has been captured by P if their distance in the plane is lower than agiven threshold ɛ > 0. Introducing z ≡ (x P , y P , x E , y E ) we can say that the captureoccurs whenever z ∈ ˜T where{˜T ≡ z ∈ R 4 : √ }(x P − x E ) 2 + (y P − y E ) 2 < ɛ . (7.169)The Isaacs equation is set in R 4 since every player belongs to R 2 . However theresult of the game just depends on the relative positions of P and E, since theirdynamics are homogeneous. In order to reduce the amount of computations neededto compute the value function we describe the game in a new coordinate systemintroducing the variables˜x = (x E − x P ) cos θ − (y E − y P ) sin θ (7.170)ỹ = (x E − x P ) sin θ − (y E − y P ) cos θ (7.171)(7.172)The new system (called relative coordinates system) has the origin fixed on theposition of P and moves with this player (see Figure 7.6.3). Note that the y axis isoriented from P to E. In the new coordinates the dynamics (7.168) becomes{˙˜x = v E sin θ E − v P sin θ P˙˜y = v E cos θ E − v P cos θ P(7.173)and the target (7.169) isT ≡{(x, y) : √ }x 2 + y 2 < ɛ .It is important to note that the above change of variables greatly simplifies thenumerical solution of the problem for three different reasons. The first is that wenow solve the Isaacs equation in R 2 and we need a grid of just M 2 nodes insteadof M 4 nodes (here M denotes the number of nodes in one dimension). The secondreason is that we now have a compact target in R 2 whereas the original target (7.169)is unbounded. This is a major advantage since we can choose a fixed rectangulardomain Q in R 2 such that it contains T and compute the solution in it. Finally, we

✐✐238 Chapter 7. Control and Gamesget rid of the boundary conditions on ∂Q. It is easily seen that the game has alwaysa value and that the only interesting case is v P > v E (if the opposite inequalityholds true capture is impossible if E plays optimally). In this situation the beststrategy for E is to run at maximal velocity in the direction opposite to P along theline passing through the initial positions of P and E. The optimal strategy for P isto run after E at maximal velocity. The corresponding minimal time of capture is√(xE − x P )T (x P , y P , x E , y E ) =2 + (y E − y P ) 2v P − v Eor, in relative coordinates,T (x, y) =√x2 + y 2v P − v E.Let us comment some numerical experiments. We have chosen Q = [−1, 1] 2v P = 2, v E = 1, A = B = [−π, π].Figures 7.12 correspond to the following discretization# Nodes ∆t ɛ # Controls23 × 23 0.05 0.20 P=41 E=41The value function is represented in the relative coordinate system, so P is fixedat the origin and the value at every point is the minimal time of capture (afterKružkov transform). As one can see in Figure 7.12, the behaviour is correct since itcorrespond to a (rescaled) distance function. The optimal trajectories for the initialpositions P = (0.3, 0.3), E = (0.6, −0.3) are represented in Figure 7.12 (right).7.6.4 The Tag-Chase game with constraints on the directionsThis game has the dynamics (7.168). The only difference with respect to the Tag-Chase game is that now the pursuer P has a constraint on his displacement directions.He can choose his control in the set θ P ∈ [a 1 , a 2 ] ⊆ [−3/4π, 3/4π]. Theevader can still choose his control as θ E ∈ [b 1 , b 2 ] = [−π, π], i.e.A = [θ 1 , θ 2 ] and B = [−π, π].In the numerical experiment below we have chosen v P = 2, v E = 1, A = [ 3 4 π, 3 4 π]and B = [−π, π]. As one can see in Figure 7.13 the time of capture at points whichare below the origin and which cannot be reached by P in a direct way have a valuebigger than at the symmetric points (above the origin). This is clearly due to thefact that P has to zig-zag to those points because the directions pointing directlyto them are not allowed as one can see in Figure 7.13 (right).7.6.5 The Homicidal chauffeurThe presentation of this game follows [FS05]. Let us consider two players (P andE) and the following dynamics:⎧ẋ P = v P sin θ,⎪⎨ ẏ P = v P cos θ,ẋ E = v E sin b,(7.174)ẏ E = v E cos b,⎪⎩ ˙θ = R v Pa,

✐✐7.6. Dynamic Programming for differential games 239Figure 7.10. Zermelo problem: boat directionysyyPPtePxyEvPEteEvExPxExsFigure 7.11. The Tag-Chase gameTest 31Test 3: P=(0.3,0.3) E=(0.6,-0.3)0.8v(x1,x2)0.80.70.60.50.40.30.20.10-0.5-1x20.60.40.20-0.2PE10.5x10-0.5-110.50x2-0.4-0.6-0.8-1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1x1Figure 7.12. Tag-Chase game: value function (relative coordinates) andoptimal trajectoriesTest 4v(x1,x2)1Test 4: P=(-0.5,0.8) E=(-0.5,0.0)0.90.80.70.60.50.40.30.20.10x20.50-0.5PE-1-0.5x200.5110.50x1-0.5-1-1-1.5-2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1x1Figure 7.13. Sector Tag-Chase game: value function and optimal trajectories

✐✐240 Chapter 7. Control and GamesysyyEyPPtevPEbvERaxPxxExsFigure 7.14. The Homicidal Chauffeur problemwhere a ∈ A ≡ [−1, 1] and b ∈ B ≡ [−π, π] are the two player’s controls. Thepursuer P is not free in his movements, he is constrained by a minimum curvatureradius R. The target is defined as in the Tag-Chase game. Also in this example wehave used the reduced coordinate system (7.173). We have considered the homicidalchauffeur game where Q = [−1, 1] 2 , v P = 1, v E = 0.5, R = 0.2 and the followingdiscretization parameters:# Nodes ∆t ɛ # Controls120 × 120 0.05 0.10 P=36 E=36Figure 7.15 shows the value function of the game. Note that when E is in front ofP the behaviour of the two players is analogous to the tag-chase game: in this case,indeed, the constraint on P ’s radius turn does not come into action (Figure 7.16(left)). However, on the P sides the value function has two higher lobes. In fact, toreach the corresponding points of the domain, the pursuer must first turn aroundhimself to be able to catch E following a straight line (see Figure 7.16 (right)).Finally, behind P there is a region where capture is impossible (v = 1) because theevader has the time to exit Q before the pursuer can catch him. Figure 7.16 (right)shows a set of optimal trajectories near a barrier in the relative coordinates system .Figure 7.17 (left) is taken from [M71] and shows the optimal trajectories which havebeen obtained by Merz via analytical methods. One can see that our results are quiteaccurate since the approximate trajectories (Figure 7.17 (right)) look very similarto the exact solutions sketched in Figure 7.17 (left). Moreover, in the numericalapproximation the barrier curve is clearly visible: that barrier cannot be crossedif both the players behave optimally. It divides the initial positions from whichthe trajectories point directly to the origin from those corresponding to trajectoriesreaching the origin after a round trip.7.7 Commented referencesAs we already mentioned, the application to deterministic control problems andgames has been one of the main motivations for the development of the theory ofviscosity solutions. Dynamic Programming (DP) singles out the value function v(x)as a key tool in the analysis of optimal control problems and zero-sum differentialgames. This technique has been introduced in the classical works on calculus of

✐✐7.7. Commented references 241Test 5v(x1,x2)10.50-1-0.5x200.5110.50x1-0.5-1Figure 7.15. Homicidal Chauffeur, value function1Test 5: P=(-0.1,-0.3) E=(0.1,0.3)1Test 5: P=(0.0,0.2) E=(0.0,-0.2)0.80.80.60.60.40.4E0.20.2Px20x20-0.2-0.2EP-0.4-0.4-0.6-0.6-0.8-0.8-1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1x1-1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1x1Figure 7.16. Homicidal Chauffeur, optimal trajectories1Test 50.80.60.40.2x20-0.2-0.4-0.6-0.8-10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x1Figure 7.17. Homicidal Chauffeur optimal trajectories: Merz description,computed trajectories

✐✐242 Chapter 7. Control and Gamesvariations and extensively applied to optimal control problems by R. Bellman. Inthe sixties, a number of discrete-time control problems were investigated via theDP method, showing that v satisfies a functional (difference) equation and thatits knowledge allows to derive optimal feedbacks. We refer to the classical books[Be87] and [BeS78] for an extended presentation of these results. In continuous-timeproblems the DP equation is a nonlinear PDE, and in principle it would be necessaryto show first that the equation has a unique solution in order to identify it withthe value function. By the time DP techniques were introduced, this was a majoranalytical difficulty, since, in general, the value functions of deterministic optimalcontrol problems and games may fail to be differentiable and even continuous, sothe classical concept of solution could not be applied. Nevertheless, Isaacs usedextensively the DP principle and the first order PDE which is now associated to hisname in his book [I65] on two-persons zero-sum differential games, working mainlyon the explicit solution of several examples in which the value function is regulareverywhere except on some smooth surfaces.Only at the beginning of the eighties, M.C. Crandall and P.L. Lions [CL83]introduced the notion of weak solution in the viscosity sense for a class of first-orderPDEs including Isaacs’ equations, and proved their existence, uniqueness and stabilityfor the main boundary value problems. The theory was reformulated by Crandall,Evans and Lions in [CEL84], and Lions proved in [Li82] that continuous valuefunctions of optimal control problems are viscosity solutions of the correspondingDP equations. Similar results were obtained for the value function of some zerosumdifferential games by Barron, Evans, Jensen and Souganidis in [BEJ84], [ES84],[Sor93a] for various definitions of upper and lower value. Independently, Subbotin[Su80, Su84] found that the value functions in the Krasovskii–Subbotin sense [KS88]of some differential games satisfy certain inequalities for the directional derivativeswhich reduce to the Isaacs’ equation at points of differentiability. Moreover, heintroduced a different notion of weak solution for first order nonlinear PDEs, theminmax solution (where the name is related to the minmax operator appearing inthe DP equation of games). The book [Su80] presents this theory with a specialemphasis on the solution of differential games which motivated his study and thename for these solutions. The equivalence of this notion of weak solution with viscositysolutions has been addressed by several authors (see [EI84, LS85, SuT86]),and currently the two theories are essentially unified [Su95].Concerning control problems with state constraints, the first characterizationof the value function is due to Soner [So86a, So86b] under a controllability assumptionswhich ensure continuity. This result has been later extended by Ishii andKoike [IK96], whereas a recent formulation due to Bokanowski, Forcadel and Zidani[BFZ10a] drops controllability assumptions still getting a characterization in termsof a DP equation. At the present state of development, the theory of viscosity solutionsapplies to a wide range of boundary and Cauchy problems for first and secondorder nonlinear PDEs, including discontinuous solutions (at least for a subclasswhich covers DP equations for control problems and games [B00, BJ89, Sor93a]).Two extensive surveys on its application to control problems and games are givenin the books [FS93] and [BCD97].We should also mention an alternative characterization of the value function,which is based on viability theory [A91]. In many classical problems, the graphof the value function is the boundary of the viability kernel for a new dynamicsassociated to the original control problem. This relationship has produced interestingresults, mainly for state-constrained problems where it is natural to consider

✐✐7.7. Commented references 243(lower) semicontinuous solutions. We refer to [Fr93] and [BJ90, BJ91] for two differentcharacterizations of semicontinuous solutions of HJB equations, and to [FP00]for an application to a state-constrained control problem.The approximation of the value function in the framework of viscosity solutionshas been investigated by several authors starting from Capuzzo Dolcetta [CD83,CDI84, F87]. Abstract convergence results, as well as a priori error estimates, havebeen proved for the approximation of classical control problems and games, see e.g.[BF90a], [BF90b], [CDF89], [Al91a], [Al91b], [BS91b]. Most of the material in thischapter originates from the survey papers [F97] and [BFS94], devoted respectivelyto control problems and games.The approximation of feedback controls is still a widely open problem, especiallyin the case of games. A first convergence result requiring regularity assumptionson the value function can be found in [DS01]. The synthesis of feedbackcontrols has also been addressed in [F01], obtaining L 1 error estimates for the approximationof optimal trajectories.Techniques for the numerical approximation of games (mainly pursuit–evasiongames) have been originated by the Krasovskii–Subbotin theory. The numericalschemes are based on a suitable construction of generalized gradients in the finitedifference operators used to approximate the value function. This approach hasbeen developed by the russian school (see [TUU95, PT00]) and allows to computean approximation of optimal feedback controls [T99]. A different technique for theapproximation of value and optimal policies of a dynamic zero-sum stochastic gamehas been proposed in [TA96], [TPA97] and is based on the approximation of thegame by a finite state approximation.The numerical counterpart of the viability approach is based on the approximationof the viability kernel. Details on the related techniques can be found in[CQS99] and [CQS00]. Finally, we mention that numerical methods based on theapproximation of open-loop controls have also been proposed, although this approachhas not been pursued here. The advantage is to replace the approximationof the DP equation (which can be difficult or impossible in high-dimensional problems)by a large system of ordinary differential equations exploiting the necessaryconditions for the optimal policy and trajectory. A general review on the relatedmethods can be found in [P94].

✐✐244 Chapter 7. Control and Games

✐✐Chapter 8Fluid DynamicsThis chapter collects some extensions of the previous algorithms to CFD applicationsas well as some examples and tests in R 2 related to more challenging benchmarks.8.1 The incompressible Euler equation in R 2The most classical model of inviscid fluid dynamics dates back to Euler and is namedafter him. In its simplest formulation, the Euler equation describes the evolutionof an inviscid, incompressible, homogeneous fluid, and takes the form of the system{u t + (u · ∇)u + ∇p = 0(8.1)∇ · u = 0,with suitable initial and boundary conditions. Here, for d ≤ 3, u = (u 1 , . . . , u d ) tdenotes the fluid velocity, and the second condition enforces mass conservation.Writing the operator (u · ∇) more explicitly, (8.1) reads⎧∂u id∑⎪⎨ ∂t + j=1d∑⎪⎩i=1∂u i∂x i= 0.u j∂u i∂x j+ ∂p∂x i= 0 (i = 1, . . . , d)*** risultati di buona posizione? proprietá qualitative? paradossi? ****** mettere il rotore (curl) tra le notazioni generali ****** notazione ∇ ↔ D? ∇ 2 ↔ ∆? ***8.1.1 Vorticity–stream function formulationAs usual in fluid dynamics, we define the vorticity as the curl of velocity:(8.2)ω = ∇ × u (8.3)In the special case of two space dimensions, the vorticity has only one nozero component(the one related to x 3 ) and can therefore be treated as a scalar. On the245

✐✐246 Chapter 8. Fluid Dynamicsother hand, once defined the operator⎛∇ ⊥ := ⎝− ∂∂x 2∂∂x 1the divergence-free vector field u may be written as⎞⎠u = −∇ ⊥ ψ (8.4)for a suitable function ψ, termed as stream function. Using (8.4) into (8.3), we get−∆ψ = ω. (8.5)Taking the curl of the momentum equation, and recalling that ∇p has a zero curl,we obtainω t + u · ∇ω = 0, (8.6)which states that vorticity is advected along the streamlines of the velocity u.Equations (8.4)–(8.6) represent the so-called vorticity–stream function formulationof the two-dimensional incompressible Euler equation.8.1.2 SL schemes for the vorticity–stream function modelThe use of SL schemes for the VS model dates back to Wiin–Nielsen, who proposedwhat is usually recognized as the first SL scheme in Numerical Weather Prediction.The scheme was originally formulated for the barotropic vorticity equation, but werewrite it here in a conceptually equivalent form for the Euler equation.The general idea is to use a SL technique to implement the advection ofvorticity in (8.6). Once vorticity is computed at a new time step, stream functionand velocity are updated via respectively (8.5) and (8.4). To keep a second-orderaccuracy in time, a midpoint scheme is used in the advective step as shown inChapter 5 (see (5.61)). More explicitly, we are given an initial condition u 0 , acorresponding stream function ψ 0 such thatu 0 = ∇ ⊥ ψ 0 ,and an initial vorticity ω 0 = ∇ × u 0 . Moreover, we assume that a further startupvalue of the vorticity ω −1 is available.A step forward in time is carried out via four substeps. First, the feet ofcharacteristics are computed as in (5.61), that is,(X ∆ (x j , t n+1 ; t n−1 ) = x j − 2∆t I[U n xj + X ∆ )(x j , t n+1 ; t n−1 )]. (8.7)2Second, an advection step for the vorticity is performed asω n+1j = I[Ω n−1 ] ( X ∆ (x j , t n+1 ; t n−1 ) ) . (8.8)Third, the updated stream function is computed by (numerically) solving the Poissonequation−∆Ψ n+1 = Ω n+1 , (8.9)

✐✐8.2. The Shallow Water Equation 247and last, the velocity is updated asU n+1 = ∇ ⊥ Ψ n+1 . (8.10)With some abuse of notation, in (8.9)–(8.10) we have used the same symbols ∆and ∇ ⊥ to denote approximate (e.g., finite difference) versions of the correspondingcontinuous operators.Note that, as it has been discussed in Chapter 5, to achieve second-orderaccuracy in time it suffices to use a P 1 or Q 1 interpolation in (8.7).8.2 The Shallow Water EquationA more complex and useful model is given by the shallow water equation (SWE),which models the flow of a relatively thin layer of fluid, with a free surface andsubject to friction at the bottom. In compact form, the system of equations reads{u t + (u · ∇)u + g∇z + γu − τ = 0(8.11)z t + ∇ · [(h + z)u] = 0.Here, u stands again for the velocity of the fluid, h denotes the depth of the fluid atthe equilibrium, and z the difference between the actual and the equilibrium depth,so that h+z is the total depth of the fluid. The bottom friction coefficient has beendenoted by γ, and has the expressionγ =g|u|C 2 z (h + z)whereas τ denotes the stress operated by external forces (typically, wind).8.2.1 The one-dimensional case: characteristic decompositionIn the particular case od a single space dimension, the SWE (8.11) takes the form{u t + uu x + gz x = −γu + τz t + (h + z)u x + uz x = −uh x .Defining the solution vector W = (u, z) t , the one-dimensional SWE may be rewrittenasW t + A(W )W x = D(W ),with A(W ), D(W ) defined by( )u gA(W ) =, D(W ) =h + z u( )−γu + τ.−uh xIn this setting, it is known that the speeds of propagation of the solution are givenby the eigenvalues of A, which are easily computed asu ± √ g(h + z),in which the terms u and ± √ g(h + z) represent respectively the speed of advectionand the motion of the gravitational waves. This latter component is usually muchfaster then the former, and causes therefore strong stability limits in conventionaleulerian schemes.

✐✐248 Chapter 8. Fluid DynamicsSL schemes based on characteristicsSemi-implicit SL schemes8.2.2 The two-dimensional case8.3 Some additional techniques8.3.1 Conservative SL schemesWhen applied to fluid dynamics problems, a major drawback of SL schemes is thatthey are not strictly mass conserving. This is due to their structure, which is notoriginally in conservative form, nor can be recast as such. In order to overcomethis problem, and try to put together the advantages of both approaches, flux-formversions of SL schemes have been proposed starting from the mid-90s.We briefly outline the underlying idea, working on the simplified model of theconstant-coefficient advection equation on R, which will be formally put in the formof a continuity equation:u t + (cu) x = 0. (8.12)First, as usual in conservative schemes, we divide the x-axis in cells of the form[x j−1/2 , x j+1/2 ] around x j , as shown in Fig. ??. A numerical solution V n representsan approximation of the cell averages of the solution at time t n , so thatvj n ≈ 1 ∫ xj+1/2u(x, t n )dx,∆x x j−1/2and, in particular, the value vj 0 is the average over the j-th cell of the initial conditionu 0 . We also need a reconstruction operator R[V n ](x) to compute the numericalsolution V n at a generic x. The reconstruction operator R is assumed to preservethe cell averages of V n , and in the simplest case, it is implemented by taking thevalue constant and equal to vjn on the whole cell.In order to update the cell averages at the time step n + 1, we start from themass balance of the j-th cell. Assuming for example that c > 0, the mass enteringthe cell from the interface x j−1/2 in the time step [t n , t n+1 ] is∫ xj−1/2x ∗ j−1/2v(x, t n )dx,where x ∗ j−1/2= x j−1/2 − c∆t denotes the foot of the characteristic starting at(x j−1/2 , t n+1 ). Applying this general idea to a numerical solution, we define anumerical flux by averaging over a time step the mass of numerical solution crossingthe interface:F j−1/2 (V n ) := 1 ∆t∫ xj−1/2x ∗ j−1/2R[V n ](x)dx.With this definition, a conservative version of the SL scheme can be defined asv n+1j= vj n + ∆t [Fj−1/2 (V n ) − F j+1/2 (V n ) ]∆x

✐✐8.4. Numerical Tests 2498.3.2 Treatment of diffusive termsTo sketch the basic ideas, we consider here the constant-coefficient advection–diffusion equation on R:u t + cu x − νu xx = 0. (8.13)We claim that (8.13) can be approximated via the schemev n+1j = 1 (2 I[V n ] x j − c∆t − √ )2ν∆t + 1 (2 I[V n ] x j − c∆t + √ )2ν∆t . (8.14)First,√note that (8.14) performs an average of two different upwind points x j −c∆t±2ν∆t. This clearly preserves the stability properties of the inviscid version, e.g.,if the interpolation I[·] is monotone, then (8.14) is also monotone. To show that itis consistent, we neglect for simplicity the interpolation step, and writeu(x j − c∆t ± √ )(2ν∆t, t n = u(x j , t n ) + −c∆t ± √ )2ν∆t u x (x j , t n ) ++ 1 (−c∆t ± √ 22ν∆t)uxx (x j , t n ) +2!+ 1 (−c∆t ± √ 32ν∆t)uxxx (x j , t n ) + O(∆t 2 ) =3!(= u(x j , t n ) + −c∆t ± √ )2ν∆t u x (x j , t n ) ++ 1 (2ν∆t ∓ 2c∆t √ )2ν∆t u xx (x j , t n ) +2!+ 1 (± √ 2ν∆t 3) u xxx (x j , t n ) + O(∆t 2 ).3!We have therefore1(2 u x j − c∆t + √ )2ν∆t, t n + 1 (2 u x j − c∆t − √ )2ν∆t, t n == u(x j , t n ) + ∆t [ − cu x (x j , t n ) + νu xx (x j , t n ) ] + O(∆t 2 ).Recalling that the interpolation error is in general O(∆x r+1 ) for a smooth solution,so thatu(x j − c∆t ± √ ) (2ν∆t, t n = I[U n ] x j − c∆t ± √ )2ν∆t + O(∆x r+1 ),we can conclude that the consistency error is of order O(∆t) + O(∆x r+1 /∆t).8.4 Numerical Tests8.4.1 Linear advection8.4.2 Examples in REsempi da [FFM01] sull’ordine di convergenza e la localizzazione dell’errore (?)8.4.3 Examples in R 2Rotating cone

✐✐250 Chapter 8. Fluid Dynamics8.5 Commented referencesMarchioro–Pulvirenti (ref. generale sull’eq. di Eulero)Letteratura su SISLNavier–Stokes: Karniadakis, Milstein–TretyakovDiffusione nonlineare: TeixeiraSL conservativiVarie su problemi del secondo ordine

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✐✐Indexadvection–diffusion equation, 249SL scheme, 249amplification factor, 90autocorrelation integral, 153Barles–Souganidis theorem, 93barotropic vorticity equation, 246Bellman equation, 191boundary conditions, 36Dirichlet, 36, 125, 139Neumann, 36, 139periodic, 125, 139state constraints, 36, 140boundary layer effect, 18change of topology, 5circulating matrices, 90coercivity condition, 29comparison principle, 26conservation, 248consistency, 79, 84generalized form, 94of multistep schemes, 46of one-step schemes, 44of the Courant–Isaacson–Rees scheme,115of the high-order SL scheme, 150,178of the Lax–Friedrichs scheme, 109,131of the SL scheme, 133, 134of the upwind scheme, 105, 129constructionof the Courant–Isaacson–Rees scheme,113of the first-order SL scheme, 133of the high-order SL scheme, 149,177of the Lax–Friedrichs scheme, 109,131of the upwind scheme, 104, 129convergence, 81, 84of multistep schemes, 47of one-step schemes, 45of the Courant–Isaacson–Rees scheme,117of the first-order SL scheme, 138of the high-order SL scheme, 172,188of the Lax–Friedrichs scheme, 112,132of the upwind scheme, 107, 131cost functionalinfinite horizon problem, 193Courant–Friedrichs–Lewy (CFL) condition,85Courant–Isaacson–Rees scheme, 113,144Crandall–Lions theorem, 92differenced form, 92differential games, 191, 228non-anticipating strategies, 231pursuit-evasion games, 228diffusive schemes, 101discount factor, 193dispersive schemes, 101distance function, 7domain of dependence, 85dynamic programming, 6eikonal equation, 7ENO interpolationin R 1 , 52in R d , 64feedback controls, 193finite elementdefinition, 66finite element interpolation, 64fixed-point iteration, 83flux-form SL schemes, 248265

✐✐266 IndexformulaHopf–Lax, 28, 30front propagation, 4Galerkin projection, 150Gibb’s oscillations, 48, 52gravitational waves, 247Hölder norms, 78image irradiance equation, 10interpolation errorfor ENO interpolation, 54for finite element interpolation, 68for Lagrange interpolation, 52for WENO interpolation, 62invarianceby translation, 88with respect to the addition ofconstants, 86Isaacs equation, 191Kružkov transformation, 202, 211, 231Lagrange finite elements, 66Lagrange interpolationin R 1 , 48in R d , 63Lagrange–Galerkin schemes, 150Lax–Friedrichs scheme, 108, 131, 144Lax–Richtmeyer theorem, 81Legendre transform, 28level set method, 4Lin–Tadmor theorem, 98linear weights, 56Lipschitz stability, 178marginal functions, 32matrix norms, 78merging, 5methodlevel set, 4of characteristics, 28of lines, 104, 109minimizationdescent methods, 71direct search methods, 70Powell method, 72trust-region methods, 73modified equation, 101of the Courant–Isaacson–Rees scheme,119of the high-order SL scheme, 173of the Lax–Friedrichs scheme, 112of the upwind scheme, 107monotonicity, 87and nonexpansivity in ∞–norm,88and nonexpansivity in Lipschitznorm, 88generalized form, 94of the Courant–Isaacson–Rees scheme,116of the first-order SL scheme, 137of the Lax–Friedrichs scheme, 111,132of the upwind scheme, 106, 130narrow band, 6normal matrix, 89numerical dispersionof high-order SL schemes, 173numerical flux, 248numerical Hamiltonian, 92numerical viscosity, 101of the Courant–Isaacson–Rees scheme,118of the Lax–Friedrichs scheme, 112of the upwind scheme, 107open-loop controls, 193optimal control, 6, 191concatenation property, 195Dynamic Programming Principle,193feedback map, 214finite horizon problem, 6, 197infinite horizon problem, 193Linear Quadratic Regulator problem,196minimum time function, 199minimum time problem, 199optimal stopping problem, 198Pontryagin Maximum Principle,193synthesis of feedback controls, 214target set, 199value function, 6Ordinary Differential Equationsapproximation, 43

✐✐Index 267multistep schemes, 45, 122one-step schemes, 44, 120Pontryagin Maximum Principle, 202positive semi-definite functions, 154problemminimum time, 7Shape-from-Shading, 9proper representation, 4WENO interpolationin R 1 , 55in R d , 64reachable set, 199reference basis functions, 48, 49, 51representation formulae, 28Riccati equation, 196running cost, 193semi-concave stability, 98discrete formulation, 100semiconcavity, 32, 177shallow water equation, 247small time local controllability, 200smoothness indicators, 56stability, 80of multistep schemes, 46of one-step schemes, 45of the Courant–Isaacson–Rees scheme,116of the high-order SL scheme, 150,178of the Lagrange–Galerkin scheme,151of the Lax–Friedrichs scheme, 110of the upwind scheme, 106sufficient conditions, 80stationary advection equation, 142stream function, 246sub and super-differentials, 24system of base characteristics, 2time-marching schemes, 83, 142, 204upwind scheme, 104, 129, 143value function, 193, 196, 198, 199, 202,205, 209viscosity solution, 18, 25viscosity solutionsfor the evolutive case, 27vorticity, 245vorticity–stream function formulation,246