A New Crank-Nicolson-Like ALE Finite Volume Scheme Verifying ...

Conference on Modelling Fluid Flow (CMFF’03)
The 12 th International Conference on Fluid Flow Technologies
Budapest, Hungary, September 3 - 6, 2003
A NEW CRANCK-NICOLSON-LIKE ALE FINITE VOLUME SCHEME VERIFYING THE
DISCRETE GEOMETRIC CONSERVATION LAW FOR 3D FLOW COMPUTATIONS ON
MOVING UNSTRUCTURED MESHES
Ingrid LEPOT 1 , Didier VIGNERON 2 and Jean-André ESSERS 3
University of Liège, ASMA Department
Institut de Mécanique et Génie Civil (Bât B52/3)
1, Chemin des Chevreuils – B-4000 Liège, Belgium
1
FNRS Research Fellow
Aerodynamic Group
2
FRIA Research Fellow
Aerodynamic Group and
Turbomachinery Group
3
Professor
Aerodynamic Group
ABSTRACT
This paper describes a finite volume solver for
the computation of unsteady Euler flows on moving
unstructured grids. The Arbitrary Lagrangian
Eulerian formulation takes the grid velocity into
account. A Cranck-Nicolson time integration
scheme and a quadratric spatial reconstruction
technique are applied and yield a second order time
and space truncation errors. A new way of satisfying
the Discrete Geometric Conservation Law adapted
to this scheme is proposed. This new method is
computational efficient as it leaves the number of
fluxes evaluations equal to their fixed grid
counterpart. Results highlighting the robustness and
accuracy of the proposed approach are presented.
Key Words: discrete geometric conservation
law, high order finite volume scheme, moving
unstructured grids.
NOMENCLATURE
h [ m ] characteristic grid size
J
2
[ m ] jacobian
M ∞
[-] Mach number
n [-] unitary normal vector
t [s] time
u [ ] conservative variables vector
v G
[m/s] grid velocity vector
x [m] spatial coordinates
F [ ] advective fluxes vector
R [ ] spatial discretized operator
T [ ] grid vibration period
α [-] time parameter
ε [-] relative error
ξ
[-] surface integration parameter
η
[-] surface integration parameter
μ
2
[ m ] non unitary normal
ω [1/s] grid vibration pulsation
∆ t [s] numerical time step
θ [-] time integration scheme parameter
2
Σ [ m ]
3
Ω [ m ]
cell surface
cell volume
1. INTRODUCTION
In many unsteady fluid mechanics applications,
the fluid movement is induced by the boundary
movement and the flow problem is necessarily
solved on a deforming grid. In those cases, the
Euler equations must be written in Arbitrary
Lagrangian Eulerian (ALE) formulation taking the
grid velocity into account. The semi-discrete finite
volume form of these equations is as follows:
d
d
dt
∫ u Ω
Ω ( t)
∫
( ) ( )
+ ⎡ ⋅ − ⋅ ⎤ dΣ
=
Σ ( t)
⎣F u n u vG
n ⎦ 0
(1)
where u collects the conservative variables at the

different mesh nodes, Ω(t) are the cell volumes,
F(u).n the normal advective fluxes and v G
.n the
normal grid velocity. The time integration of these
semi-discrete equations raises the issue of how to
evaluate the normal fluxes integral on moving
surfaces.
An answer to this question issued in the early
days of Computational Fluid Dynamics (CFD) [1]
by considering the Geometric Conservation Law
(GCL) stated below
d
d
d
dt
∫ Ω −
Ω ( t) ∫ v ⋅ Σ =
Σ ( t)
G
n 0
(2)
An exact time integration turns this relation to
the continuous formulation of GCL
n 1
t
n 1 n
Ω + +
− Ω = ∫ dΣ
n
t ∫ v ⋅
Σ ( t)
G
n
(3)
This formula expresses that the change in
volume between t n and t n+1 is equal to the volume
swept by its boundary during the time step Δt = t n+1 -
t n . During their deformation, the control volumes
always respect this relation and, hence, the Euler
equations in ALE formulation has the natural
property of accepting a solution u≠0 uniform in
space and constant in time.
After space and time discretization, this
particular property is not necessarily preserved. A
solution for obtaining a new numerical scheme for
moving grid that respects this property from a wellknown
classical scheme designed for fixed grid
computations is to evaluate the geometric
parameters, such as cell volume and grid velocity, in
order to preserve a constant uniform solution at
each time step. Therefore an adequate discrete
version of the GCL (DGCL), depending not only on
the semi-discretization but also on the time
integration procedure, has to be derived for each
numerical scheme. This DGCL constitutes a
guideline to determine the geometrical quantities in
the normal fluxes integrals.
Violating the DGCL introduces spurious
oscillations affecting the accuracy of the numerical
scheme but the real necessity of the DGCL has long
remained hazy. Guillard and Farhat [2] have shown
that the DGCL enforcement is close to a time
integration consistency condition. More recently,
the relation of the DGCL enforcement to linear [3]
and nonlinear [4] stability has also been
investigated, assessing its practical importance.
Several ways of imposing the DGCL condition
for each numerical time step are possible. The GCL
can be regarded as an additional conservation law
[5] that has to be solved numerically using the same
dt
scheme that is used to integrate the fluid equations
to provide a self-consistent solution for the local
cell volumes. However, a more computationally
efficient way is to avoid solving an additional
equation and to adapt the numerical scheme with
specific geometrical quantities to ensure the DGCL
is not violated by integrating the physical quantities
conservation equations.
In this paper, the vertices movement considered
between two mesh configurations is assumed to be
linear in time, which yields a constant mesh
velocity per time interval. The spatial discretization
of the advective terms is first outlined. The implicit
geometrically conservative time integration
procedure is then described in detail. Finally,
preliminary results assessing the accuracy and
robustness of the proposed approach are presented.
2. SPATIAL DISCRETIZATION
This cell-centered finite volume solver is
designed for unstructured meshes whose volume
faces may be either triangles or (possibly non
planar) quadrilaterals, allowing among others for
tetrahedral, hexahedra, wedges or pyramids as
control volumes. The design of high order schemes
requires two elements : an accurate interpolation (or
reconstruction) of the advective fluxes from the cell
centers to the cell faces, and an accurate surface
integration of the normal fluxes. Once the
reconstruction has been performed, Roe’s flux
difference splitting is used to compute the upwinded
advective normal fluxes.
The constant reconstruction scheme (often
called first-order) only uses the left and right
neighbours values on each face and can be proved
to be inconsistent on irregular meshes. The
consistent scheme needs a linear interpolation of the
fluxes or variables to the face’s Gauss points. A
third order method, which leads to a second order
truncation error for the advective derivatives, is
obtained with a quadratic reconstruction. It requires
the calculation of centered second-order accurate
first derivatives and first-order accurate second
derivatives.
The key component of high order schemes is
the computation of the variables derivatives, used to
perform the reconstruction, at the cell centers. A
fixed stencil is selected for each node, which is the
set of neighbouring nodes whose variables values
are linearly combined to provide the
aforementioned centered derivatives. This
combination is defined in such a way that the
difference between the reconstructed values at the
location of the stencil nodes and their actual values

is minimized in a least square sense [6,7]. For the
deforming meshes considered in this work, the
derivative stencils remain constant throughout the
computation, while the weights are recomputed for
each new mesh configuration, i.e. at each time step.
If each surface point x is represented by the two
parameters ξ and η, the unitary normal n and the
jacobian matrix determinant J can be expressed as
x′ ξ
∧ x′
η
n =
x′ ′
ξ
∧ xη
(4)
J = x′ ′
ξ
∧ xη
(5)
and the advective fluxes surface integration
becomes
∫ ⎡ ( ) ( ) d d
Σ ( t)
⎣F u − u v ⎤⎦
⋅ ′ ′
G
xξ
∧ xη
ξ η
(6)
For convenience, we define below the non unitary
normal as
μ = x′ ′
ξ
∧ xη = n dΣ
(7)
Those integrals are performed by a Gauss
quadrature. Table 1 summarizes the minimum
number of Gauss points per face necessary not to
spoil the accuracy of the reconstruction procedure
on static grids.
Table 1. Number of face Gauss points on static grid
Triangles Quadrangles
Constant reconstruction 1 1
Linear reconstruction 1 4
Quadratic reconstruction 3 4
On dynamic grids, the additional ALE term,
composed of the reconstructed variables multiplied
by the normal grid velocities which are at most biquadratic
functions of (ξ,η), causes the necessary
number of Gauss points to be increased in order to
preserve the accuracy of the spatial discretization
and to respect the DGCL condition.
Table 2. Number of face Gauss points on dynamic
grid
Triangles Quadrangles
Constant reconstruction 1 4
Linear reconstruction 3 4
Quadratic reconstruction 4 9
3. TIME INTEGRATION
Defining R as the Euler equations spatial terms
( , ,
G
⋅ n)
=
⎡F ( u) n u ( v n)
R u n v
∫
− ⋅ − ⋅ ⎤ dΣ
Σ ( t)
⎣
G ⎦
(8)
the semi-discrete form of Eq. (1) can be written as
d
( u Ω ) = R ( u, n, vG
⋅n,
dΣ
)
dt
(9)
The time integration of this equation is
performed by a one-step semi-implicit scheme
leading to the following nonlinear equations system
for the unknown variables u n+1
n+ 1 n+
1 n n
u Ω − u Ω
n+
1
= θ R + ( 1−θ
)
n
∆t
(10)
Those nonlinear equations are solved by means
of a Newton iterative process at each time step and
the resulting linear system is solved by a matrixfree
GMRES algorithm. The preconditioning of the
linear system is critical to achieve a good
preformance of the Krylov subspace solvers. Hence,
a right-preconditioning of the GMRES, based on an
cheap approximation of the jacobian matrix
obtained by assuming a constant reconstruction
scheme and an Incomplete LU decomposition
solver (ILU), has been implemented.
Such a scheme yields second order accuracy in
time for θ = 0.5 ; this one-step trapezoidal rule (or
Cranck-Nicolson scheme) appears as a natural
optimum method since it is the most accurate A-
stable linear multi-step method. Hence, thanks to
the implicit treatment of the equations, the time step
size is not restricted by a stability condition and can
be as large as the smallest characteristic physical
time of the fluid flow. It has been demonstrated in
[4] that an implicit scheme violating the DGCL can
exhibit the same nonlinear stability behaviour either
on moving or fixed grid, if the time step and the
mesh distortion are small enough. In our case, the
DGCL is a crucial stability condition since the time
steps could be quite large.
Deriving the corresponding DGCL for this
particular scheme is done by imposing a constant
uniform solution u≠0 in Eq. (9). In this case, the
integration of the normal fluxes F(u).n over surface
cell volume is equal to zero and Eq. (9) yields
n+
1 n
Ω − Ω
n
∆t
= ∑ ∑
⎡θ
G
⋅ + −
G
⋅
⎣
faces Gauss
points
R
n+
1
( vμ ) ( 1 θv) ( μ )
(11)
Using the relation of Eq. (3), the latter equation
turns to
n
n
⎤
⎦

n+
1
t
∫ n
t ∫ v
Σ ( t)
G
∑ ∑
faces Gauss
points
⋅n
dΣ
dt
n+
1
( vμ ) ( 1 θv) ( μ )
= ⎡θ
G
⋅ + −
G
⋅
⎣
(12)
The right hand side (RHS) of Eq. (12) clearly
appears as a discretized approximation of the left
hand side (LHS). However that equation could be
exact depending on the way the product v G
.n
depends on time. The latter requirement should be
met for the DGCL to be satisfied. Since the vertices
coordinates are considered to vary linearly in time
during a time step, the grid velocity is constant and
the non unitary normal is quadratic for 3D meshes
and therefore the product v G
.n is also quadratic. As
the θ-scheme does not provide an exact integration
of a quadratic function even for θ = 0.5 , the non
modify Cranck-Nicolson scheme described in Eq.
(10) does not meet the DGCL for 3D moving
meshes.
A straightforward possibility to enforce the
DGCL is to evaluate both R n and R n+1 on each of the
intermediate configurations and take their weighted
average. Such a procedure however considerably
raises the computational burden by doubling the
number of normal fluxes computations.
A more computationally efficient procedure [4]
takes advantage of the fact that the normal fluxes
become linear with respect of the normal for a
uniform flow. Thanks to this property, one can
choose the geometric parameters for each R n and
R n+1 as to respect the DGCL. The distinction
between the mean normal procedure and the distinct
normals procedure is made below.
Mean normal procedure
An identical n for both normal fluxes, equal to
the arithmetical mean between the normals at the
intermediate configurations can be used as it has
been proposed in [4]. Those intermediate
configurations are taken at time t * and t **
corresponding to the Gauss points positions on the
interval time step Δt :
* n 1 3
t t ⎛ ⎞
= + 1− ∆t
2 ⎜
3 ⎟
⎝ ⎠ (13)
** n 1 3
t t ⎛ ⎞
= + 1+ ∆t
2 ⎜
3 ⎟
⎝ ⎠ (14)
This mean normal procedure leads to the
following time discretization scheme
n
⎤
⎦
u
Ω − u
n
∆t
Ω
n+ 1 n+
1 n n
n+
θ R ( u 1 , n, vG
n,
dΣ
)
n
( 1 θ ) R ( u , n, vG
n,
dΣ
)
= ⋅
+ − ⋅
(15)
where the mean geometric parameters are computed
as
1 * **
n = 2 ⎡ ⎣n + n ⎤ ⎦
(16)
1
* **
v ⋅ = ( ) ( )
2 ⎡ ⋅ + ⋅
⎣
⎤
G
n vG n vG
n
⎦
(17)
1 * **
dΣ = ⎡( dΣ ) + ( dΣ
) ⎤
2 ⎣
⎦
(18)
The DGCL is then satisfied independently of
the θ parameter. Such a normal no longer
corresponds to a physical intermediate
configuration as the face normals defined in Eq. (4)
are nonlinear functions of the vertices coordinates.
Distinct normals procedure
A new way of enforcing the DGCL is now
proposed. Integrating the product of two functions
ψ(t) and δ(t) can be done by the following time
integration method
n+
1
1 t
ψ
n
( t ) δ ( t ) dt
∆t
∫t
* n
** n+
1
( 1 θ ) ψ ( t ) δ ( t ) θψ ( t ) δ ( t )
= − +
( t )
+ O ∆
with the intermediate times t * and t ** chosen as
* n *
t = t + α ∆t
** n **
t = t + α ∆t
(19)
(20)
(21)
This scheme is generally first order accurate in
time. However, if θ=0.5 and α * +α ** =1, the second
order of accuracy of the trapezoidal rule is
recovered. In the following, we require formula in
Eq. (19) to be exact if the product ψ(t) δ(t) is a
quadratic function of time. This requirement is met
by choosing the α * and α ** as follows
⎡
⎤
*
α =
1
2
⎢1
−
⎣
3
θ
( 1−θ
) ⎥ ⎦
(22)
⎡ ⎤
** 1 1−θ
α = ⎢1
+ ⎥
2 ⎣ 3θ
⎦
(23)
To ensure that α * and α ** , which now depend on
the θ parameter, belong to the interval [0,1], θ must

e chosen between 0.25 and 0.75. For the Cranck-
Nicolson scheme (θ=0.5), we naturally recover the
Gauss points positions expressed in “Eqs. (13) and
(14)”.
Since the ALE term of Eq. (8) appears as a
product of two functions, the ψ(t) function is
assumed to contain the geometrical part such as the
grid velocity (considered constant during each time
step) multiplied by the non unitary normal μ while
the δ(t) function contains the conservative variables
part u. For 3D flow computations, the non unitary
normal is a quadratic function of time while u is
considered constant while checking the DGCL
condition. Hence, the resulting product ψ(t) δ(t) is a
quadratic function of time and the formula in Eq.
(19) is therefore exact. As a result the scheme below
satisfies the DGCL.
n+ 1 n+
1 n n
u Ω − u Ω
n
∆t
n+
**
( ( )
)
1 , ** , , dΣ
**
G
= θ R u n v ⋅n
*
( )
* dΣ
*
G
n
( θ ) R u n ( v n)
+ 1 − , , ⋅ ,
(24)
This procedure, like the mean normal
procedure, leaves the number of fluxes evaluations
equal to their fixed grid counterpart. However, the
distinct normals used for the normal fluxes
evaluations now correspond to a physical
intermediate configuration on the considered time
interval. This allows to take the positions of the face
Gauss points at these configurations and, hence, this
procedure could also be applied when the spatial
coordinates explicitly appear in the fluxes
expressions (e.g. Euler equations in cylindrical
coordinates), which can’t be handled by mean
normal procedure.
4. RESULTS
In this section, numerical results are presented.
Starting with a uniform mesh with grid size h, we
require that grid to vibrate randomly for each test
case. This leads to a completely distorded mesh a
each time step. Each vertex vibrates around its
initial position with an amplitude Δx 0
, randomly
chosen in amplitude and direction, and its position
at time t n can be calculated as
n 0 0 n
x = x + ∆x
sin ( ω t )
(25)
where the pulsation ω can be chosen.
To make sure that the proposed schemes really
meet the DGCL, a first test case has been computed
solving the Euler equations for a uniform flow on a
3D mesh.
The time accuracy is then analysed on a simple
linear scalar hyperbolic equation with a known
analytical solution for different mesh movements.
For these two test cases, comparison is made
between the normal mean procedure, the distinct
normals procedure and the non modified Cranck-
Nicolson scheme which takes all the geometric
parameters in t n and t n+1 .
Finally, some convergence properties of the
distinct normal approach are outlined on a fluid
flow in a channel with a bump.
4.1. Uniform flow
This first test case consists in a uniform flow
(M ∞
=0.2) through a rectangular duct. The goal is to
check if the DGCL is satisfied and to measure the
time evolution of the error for non-DGCL schemes.
The grid is required to vibrate with a fixed pulsation
ω=500π and several distorsion amplitudes. The
numerical time step Δt is constant and is such that
the mesh movement period is discretized on 16
points.
Realtive error
2,51E-04
2,01E-04
1,51E-04
1,01E-04
5,10E-05
dx0 = 0.05 h
dx0 = 0.1 h
dx0 = 0.15 h
1,00E-06
2,50E-04 2,00E-03 3,75E-03 5,50E-03 7,25E-03
Time t (s)
Figure 1. Time evolution of the relative error for
the non respective DGCL Cranck-Nicolson scheme
for different mesh distosion amplitudes.

Relative error
1,82E-12
1,62E-12
1,42E-12
1,22E-12
1,02E-12
8,20E-13
6,20E-13
4,20E-13
2,20E-13
Distinct normals procedure
Normal mean procedure
2,00E-14
2,50E-04 1,75E-03 3,25E-03 4,75E-03 6,25E-03
Time t (s)
Figure 2. Time evolution of the relative error for
both the distinct normals and the normal mean
approach for a mesh distorsion amplitude of dx0 =
1.5 h.
The relative error ε occurring at each time step
is computed with the following formula
ε =
∑
i
( ui
− u )
∑ ( ui
)
i
2
i
2
(26)
with u u
i and
i respectively denoting the numerical
solution and the analytical solution.
Figure 1 shows the evolution of error with time
for the non modified Cranck-Nicolson scheme
while Figure 2 shows it for both normal mean and
distinct normals approaches.
These examples clearly show that, as opposed
to the non-DGCL Cranck-Nicolson scheme, either
the normal mean and the distinct normals
techniques lead to a DGCL scheme.
4.2. Advection scalar test
The accuracy analysis is here made by solving
the following simple linear scalar hyperbolic
conservation equation on a cubic domain.
∂u
∂u
+ = 0
∂t
∂x
(27)
The initial condition consists in the continuous
spherical field
2⎛
1 ⎞
u = 1+
cos ⎜ π r ⎟
⎝ 2 ⎠ if r < 1
(28)
u = 1
if r ≥ 1
(29)
with the radius r defined as
r =
2
2
( x − 0.3) + ( y − 0.5) + ( z − 0.5)
with R=0.25.
Error
7.50E-03
6.50E-03
5.50E-03
4.50E-03
3.50E-03
2.50E-03
1.50E-03
R
Violating DGCL scheme
Distinct normal scheme
Normal mean scheme
Fixed grid error limit
2
(30)
5.00E-04
2 4 8 16 32 64
T/(2dt)
Figure 3. Time integration error on a vibrating grid
for the respecting DGCL schemes (distinct normal
and normal mean procedures) and for the non
modified Cranck-Nicolson scheme violating the
DGCL.
The analytical solution is simply a translation of
that initial field in the x direction with a unitary
speed. For accuracy analysis, the numerical solution
u obtained at t=0.08 with a time step Δt=2.5 is
compared with the analytical solution u on each
node i for the different procedures. The relative
error ε is computed by Eq. (26).
Here, the distorsion amplitude Δx 0
is maintained
constant for each test while the vibration period T of
the grid movement goes from small to large number
of time steps. Figure 3 shows the different errors
obtained for each procedure.
This example highlights that the non modified
Cranck-Nicolson scheme, which does not meet the
DGCL, exhibits larger errors for the high frequency
mesh movements and that its behaviour depends of
how the mesh movement is time discretized. On the
other hand the two DGCL schemes are less
dependent of the grid velocity. Finally we can also
observe that the obtained error for very slow mesh
motions tends to be equal to the fixed grid error
whether the scheme is DGCL or not. This remark
means that satisfying the DGCL is really essential
when the characteristic time of the grid movement
becomes comparable or smaller that the numerical
time step.

4.3. Steady plane flow in a channel with a
bump using a 3D vibrating mesh
A steady subsonic flow (M ∞
=0.3) in a channel
with a bump is computed on a tetrahedral mesh
(19378 cells) and serves as initial field before the
mesh starts to vibrate.
Number of Krylov vectors
35
30
25
20
15
10
5
0
NM-dx0=h
DN-dx0=h
NM-dx0=0.5 h
DN-dx0=0.5 h
NM-dx0=0.25 h
DN-dx0=0.25 h
1 2 3 4 5 6
Nonlinear iterations
Figure 4. Convergence curves : number of Krylov
vectors per nonlinear Newton iterations for several
mesh distorsion amplitudes (Δx 0
= 0.25h, Δx 0
= 0.5h
and Δx 0
= h) using the normal mean procedure (MN)
and the distinct normals procedure (DN).
The only unsteadyness is here brought by the
mesh movement. Figure 4 shows the necessary
number of Krylov vectors within each nonlinear
Newton iteration until the convergence is obtained
from the numerical solution at time t n to the new
solution at time t n+1 .
For a random Δx 0
= 0.25h distorsion, both mean
normal and distinct normals approaches exhibit the
same behaviour. However, for a Δx 0
=0.5h
distorsion, the distinct normals approach shows an
improved convergence rate. This becomes more
critical for a Δx 0
=h distortion. For the latter, it
should be noted that even some mesh crossovers
appear (about 4.8% of the volumes) ; this however
doesn’t alter the conservativity of the finite volume
discretization and solutions could be obtained by
both methods on the cases presented here with such
extreme conditions.
showed an improved convergence rate with respect
to a normal mean approach, with the same accuracy.
This new technique could also be applied when the
spatial coordinates explicitly appear in the fluxes
expressions (e.g. Euler equations in cylindrical
coordinates), which can’t be handled by mean
normal procedure.
Further investigations need of course to be
carried out for unsteady flows with moving
boundaries.
REFERENCES
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“Geometrical Conservation Law and its
Applications to Flow Computations on Moving
Grids”, AIAA Journal, 17:1030, 1979.
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for Flow Computations on Moving Meshes”,
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[3] Fromaggia L., and Nobile F., “A Stability
Analysis for the Arbitrary Lagrangian Eulerian
Formulation with Finite Elements”, East-West J.
Numeric Math., 7:107, 1999.
[4] Farhat C., Geuzaine P., and Grandmont C.,
“The Discret Geometric Conservation Law and the
Nonlinear Stability of ALE Schemes for the
Solution of Flow Problems on Moving Grids”,
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[5] Batina J.T., “Unsteady Euler Airfoil
Solutions Using Unstructured Dynamic Meshes”,
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[6] Delanaye M., and YenLiu, “Quadratic
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2001.
5. CONCLUSION
A geometrically conservative high order scheme
has been presented. The importance of verifying the
Discrete Geometric Conservation Law for
computations on highly moving grids has been
outlined on simple test cases.
The new proposed DGCL enforcement has