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248 J.-M. Brun and B. Mohammadi 2.2 Long Range Transport and Non-Symmetric Geometry The modelling above gives a local distribution for the advected quantities. We are now interested by the quantities candidate for a transport over large distances. We suppose that those are given by ∫ c + (x, y) = c l dz or c + (x, y) =u + l c l , where H ∼ 2 − 3m and u + l transported is given by z>H = max(0, (u · z)/‖u‖). The total quantity being ∫ C = c + (x, y)dσ, R 2 which should be conserved by the reduced-order transport model we would like to build and for which c + is the input condition. One aims now to again reduce the search space for the solution. The primary factors influencing the dispersion of a neutral plume are advection by the wind and turbulent mixing. The simplest model of this process is to assume that the plume advects downwind and spreads out in the horizontal and vertical directions. Hence, the distribution of a passive scalar c, emitted from a given point and transported by a uniform plane flow filed U along x-coordinate, is given by where c(x, y, z) =c c (x)f( √ y 2 + z 2 ,δ(x)), (3) c c (x) ∼ exp(−a(U)x) and f( √ y 2 + z 2 ,δ(x)) ∼ exp(−b(U, δ(x)) √ y 2 + z 2 ). c c is the behavior along the central axis of the distribution and δ(x) characterizes the thickness of the distribution at a given x-coordinate. An analogy exists with plane or axisymmetric mixing layers and neutral plumes where δ is parabolic for a laminar jet and linear in turbulent cases [Cou89, Sim97]. a(·) is a positive monotonic decreasing function and b(·, ·) is positive, monotonic increasing in U and decreasing in δ. In a uniform atmospheric flow field, this solution can be used for the transport of c + above. We would like to generalize this solution in a non-symmetric metric defined by migration times based on the flow field and hence treat the case of variable flow fields. Nonsymmetric Geometry In a symmetric geometry the distance function between two points A and B verifies d(A, B) =0⇒ A = B, d(A, B) =d(B,A), d(A, B) ≤ d(A, C)+d(C, B).
Reduced-Order Modelling of Dispersion 249 But the distance function can be non-uniform with anisotropy (the unit spheres being ellipsoids). In a chosen metric M the distance between A and B is given by d M (AB) = ∫ 1 0 ( t −→ ABM(A + t −→ AB) −→ AB ) 1/2 dt, where M is positive definite and symmetric in symmetric geometries. With M = I, one recovers the Euclidean geometry and variable M permits to account for anisotropy and non-uniformity of the distance function. We have widely used this approach for mesh adaptation for steady and unsteady phenomenon [AGFM02, HM97, BGM97] linking the metric to the Hessian of the solution. This definition of the metric permits to equi-distribute the interpolation error over a given mesh and, therefore, monitor the quality of the solution. Consider now the following distance function definition: Definition 1. If A is upwind with respect to B then d(B,A) =∞ and d(A, B) = ∫ B ⊥ A ds/u = T, where T is the migration time from A to B ⊥ along the characteristic passing by A. u is the local velocity along this characteristic and is, by definition, tangent to the characteristic. B ⊥ denotes the projection of B over this characteristic in the Euclidean metric. One supposes that this characteristic is unique, hence avoiding sources and attraction points in the flow field. In case of nonuniqueness of this projection, one chooses the direction of the projection which satisfies best the constraint u ·∇c g =0inB. Generalized Plume Solution Once this distance built, we assume the distribution of a passive scalar transported by a flow u can be written as: c g = c c (d)f(d ⊥ E,δ(d)). (4) Here the subscript g reads for global and mentions long distance transport. d ⊥ E is the Euclidean distance in the normal direction local to the characteristic at B ⊥ (i.e. along direction BB ⊥ ). Flow Field One should keep in mind that in realistic configurations, one has very little information on the details of the atmospheric flow compared to the accuracy
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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50 E.J. Dean and R. Glowinski minim
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54 E.J. Dean and R. Glowinski Fig.
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62 E.J. Dean and R. Glowinski Assum
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Higher Order Time Stepping for Seco
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u n+1 h Optimal Higher Order Time D
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136 S. Lapin et al. Ω R γ Fig. 3.
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138 S. Lapin et al. 4 Energy Inequa
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140 S. Lapin et al. 5 Numerical Exp
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142 S. Lapin et al. Fig. 6. Contour
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Domain Decomposition and Electronic
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Domain Decomposition Approach for C
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Numerical Analysis of a Finite Elem
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so that |u| 2 1,ω h ≤ 2 Numerica
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