HIGH ORDER FINITE VOLUME SCHEMES

Transcription

1 HIGH ORDER FINITE VOLUME SCHEMES Jean-Pierre Croisille Laboratoire de mathématiques, UMR CNRS Univ. Paul Verlaine-Metz LMD, Jan. 26, 2011

2 Joint work with B. Courbet, F. Haider, ONERA, DSNA (Numerical Simulation of Fluid Flows: Aerodynamics & Aeroacoustics, Châtillon-sous-Bagneux, France

3 Scientific context High order numerical methods in Computational Fluid Dynamics Today, finite volume methods are commonly used in many areas of CFD: aerodynamics (external flows), aerothermochemistry(internal flows), porous media flows (petroleum industry, water resource), magnetohydrodynamics (cosmology). Main reasons: versatility of the method, good properties (treatment of convective/diffusive flows, nonlinear waves, conservativity,) strong support in the mathematical applied community. Strong support in physics AND mathematics. Strong interest for this kind of approach in the community in climatology. General context: development of the dynamical core of the GCM equations. Main model: the SW equations on the rotating spherical earth, 2D or 3D. Since several years: strong interest for this problem in the mathematical/cfd community.

4 Outline 1 A finite volume code - The package CEDRE (ONERA- DSNA): aerothermochemistry and energetics. 2 The MUSCL scheme for conservation laws 3 High order finite volumes methods 4 Perspectives: The MUSCL scheme on a spherical grid

5 The package CEDRE: Aerothermochemistry for complex fluid flows Problems to solve Full 3D Navier-Stokes equations + chemical reactions + turbulence modeling (RANS or LES) in complex geometries. 8 < : ρ t + (ρv) = 0 (ρv) t + (ρv v) + p τ) = 0 (ρe tot) t + (ρve tot + pv τ v + q th ) = 0 Fluid models Pressure law: p = p(ρ, ε). Viscous strain tensor τ(x, t) = λ L v(x, t)δ (2) + 2µ L v(x, t) (1) λ L, µ L are the Lamé coefficients, denotes the symmetric tensor product, δ (2) is the unit tensor. Large scale turbulence modeling: LES or RANS. Many other models: chemistry, diphasic flows, particles (eulerian or lagrangian), advected passive scalars.

6 The package CEDRE: Aerothermochemistry for complex fluid flows Problems to solve Full 3D Navier-Stokes equations + chemical reactions + turbulence modeling (RANS or LES) in complex geometries. 8 < : ρ t + (ρv) = 0 (ρv) t + (ρv v) + p τ) = 0 (ρe tot) t + (ρve tot + pv τ v + q th ) = 0 Fluid models Pressure law: p = p(ρ, ε). Viscous strain tensor τ(x, t) = λ L v(x, t)δ (2) + 2µ L v(x, t) (1) λ L, µ L are the Lamé coefficients, denotes the symmetric tensor product, δ (2) is the unit tensor. Large scale turbulence modeling: LES or RANS. Many other models: chemistry, diphasic flows, particles (eulerian or lagrangian), advected passive scalars.

7 Main features of CEDRE Spatial approximation General polyhedral grid Cell-centered FV method: one unknown by cell. Multidomain with parallelism Many pysical models available: single phase flows, multiphasic flows, real gas flows, chemistry, particles (eulerian or lagrangian).

8 Typical set of conservative variables Conservative variables 2 3 ρ 1 ρ nesp ρv 1 ρv 2 q = ρv 3 ρe t ρz ρz nsca physical variables 2 3 p T y 1 2 y nesp u = v 1 = v 2 v ρz ρz nsca thermo. state Velocity Scalar values: Passive scalars RANS scalars (2) 3 7 5

9 Spatial approximation: average on the cells Semi-discrete scheme Form of the semi-discrete dynamical system V i q i = X A i,j f n,i j X A ij ϕ n,i j + V i σ i (3) j V j V Upwinded Euler flux f n,i j = f n(u lim K i, u lim K j ) = 8 < : Roe Flux Low Mach (Turkel) AUSM (4) Navier-Stokes diffusive flux ϕ n,i j = ϕ n(u K, u K ) Sources σ i = σ i (u i, u i ).

10 Specificities in CEDRE (B. Courbet,DSNA-ONERA) Specificities 1 Notion of boundary cells supporting a differential equation. Plays the same role than an internal cell. Useful for multidomains. Useful for boundary conditions. The boundary conditions using relaxation differential equations supported in boundary cells. 2 A geometric domain is multisolver. Coupling of different systems of equations inside a single geometric domain.

11 Time algorithms Explicit time-schemes Runge-Kutta time schemes. Order 2,3,4. Implicit time-schemes Implicit linearized Euler scheme to compute asymptotic stationary states. Linear solvers: GMRES with digonal preconditionning. Implicit time-dependent schemes 2 or 3 implicit steps. Use of the approximate Jacobian operator based on the first order spatial scheme. Low Mach number flows: CFL hydro = 1 for the advective waves (shocks), CFL acc = for the acoustic waves.

12 Time algorithms Explicit time-schemes Runge-Kutta time schemes. Order 2,3,4. Implicit time-schemes Implicit linearized Euler scheme to compute asymptotic stationary states. Linear solvers: GMRES with digonal preconditionning. Implicit time-dependent schemes 2 or 3 implicit steps. Use of the approximate Jacobian operator based on the first order spatial scheme. Low Mach number flows: CFL hydro = 1 for the advective waves (shocks), CFL acc = for the acoustic waves.

13 Main steps of the algorithm Basic scheme The scheme reads U n+1 i = U n i t T i X A ij F(Ũi n, Ũn j ) (5) Given the values q i in the cells, interpolate the gradient q i in each cells. Compute the value of the piecewise linear function ij U n j (x) = Un j + Un j (x x j) (6) Evaluate the numerical flux-function F(Ũi n, Ũn j ). (Slope limiters are used). Assemble the contribution of each interface A ij to the cells T i and the cell T j. Update U n i by U n+1 i. Additional features The computing algorithm is much more complicated than expected at first glance! Loop on the faces, indirection problems, efficiency problems, interdomain problems. Accuracy of quadrature formulas on cell interfaces.

14 Main steps of the algorithm Basic scheme The scheme reads U n+1 i = U n i t T i X A ij F(Ũi n, Ũn j ) (5) Given the values q i in the cells, interpolate the gradient q i in each cells. Compute the value of the piecewise linear function ij U n j (x) = Un j + Un j (x x j) (6) Evaluate the numerical flux-function F(Ũi n, Ũn j ). (Slope limiters are used). Assemble the contribution of each interface A ij to the cells T i and the cell T j. Update U n i by U n+1 i. Additional features The computing algorithm is much more complicated than expected at first glance! Loop on the faces, indirection problems, efficiency problems, interdomain problems. Accuracy of quadrature formulas on cell interfaces.

15 Simulations in aerothermochemistry with CEDRE

16 Local geometry of a cell in CEDRE

17 The MUSCL method in space Conservation law tu (x, t) + f (u(x, t)) = 0 (7) Local geometry The cell with number α is denoted T α, with barycenter x α and d-volume T α. The face A αβ, with barycenter x αβ, has a normal vector n αβ oriented from cell T α to T β and of length nαβ equal to the surface Aαβ. The oriented normal unit vector of the face A αβ is ν αβ. Geometric notational convention The following convention simplifies the notation of sums over cells. Whenever two cells have no common interface, n αβ = 0 and k αβ = 0 and the face A αβ is defined to be empty so that any surface integral over A αβ is automatically zero. In addition, n αα, k αα and h αα are defined to be zero.

18 The MUSCL method in space Conservation law tu (x, t) + f (u(x, t)) = 0 (7) Local geometry The cell with number α is denoted T α, with barycenter x α and d-volume T α. The face A αβ, with barycenter x αβ, has a normal vector n αβ oriented from cell T α to T β and of length nαβ equal to the surface Aαβ. The oriented normal unit vector of the face A αβ is ν αβ. Geometric notational convention The following convention simplifies the notation of sums over cells. Whenever two cells have no common interface, n αβ = 0 and k αβ = 0 and the face A αβ is defined to be empty so that any surface integral over A αβ is automatically zero. In addition, n αα, k αα and h αα are defined to be zero.

19 The MUSCL method in space Conservation law tu (x, t) + f (u(x, t)) = 0 (7) Local geometry The cell with number α is denoted T α, with barycenter x α and d-volume T α. The face A αβ, with barycenter x αβ, has a normal vector n αβ oriented from cell T α to T β and of length nαβ equal to the surface Aαβ. The oriented normal unit vector of the face A αβ is ν αβ. Geometric notational convention The following convention simplifies the notation of sums over cells. Whenever two cells have no common interface, n αβ = 0 and k αβ = 0 and the face A αβ is defined to be empty so that any surface integral over A αβ is automatically zero. In addition, n αα, k αα and h αα are defined to be zero.

20 The MUSCL method in space, (cont.) MUSCL with Method Of Lines The simplest finite-volume scheme consists in evolving the quantities u α (t) approximating the exact averages ū α (t) with the dynamical system du α (t) = 1 X Z fαβ `uα (t), u β (t) dσ (8) dt T α β A αβ MUSCL dynamical system The numerical flux function f αβ (w int, w ext) depends on the two states w int and w ext on each side of the cell interface. Conservation is translates as f αβ (w int, w ext) = f βα (w ext, w int ) (9)

21 The MUSCL method in space, (cont.) MUSCL with Method Of Lines The simplest finite-volume scheme consists in evolving the quantities u α (t) approximating the exact averages ū α (t) with the dynamical system du α (t) = 1 X Z fαβ `uα (t), u β (t) dσ (8) dt T α β A αβ MUSCL dynamical system The numerical flux function f αβ (w int, w ext) depends on the two states w int and w ext on each side of the cell interface. Conservation is translates as f αβ (w int, w ext) = f βα (w ext, w int ) (9)

22 The MUSCL method in space, (cont.) Final form of the MUSCL scheme du α (t) = dt 1 X X ω q f αβ `wα [u(t)] `x αβ;q, wβ [u(t)] `x αβ;q. T α β q (10) Accuracy In equation (10), the x αβ;q are the quadrature points on A αβ and the ω q are the quadrature weights. If the expression under the integral on the right hand side of (8) is a polynomial of degree p in x and if the quadrature formula integrates exactly such polynomials, this step does not introduce any new discretization errors.

23 The MUSCL method in space, (cont.) Final form of the MUSCL scheme du α (t) = dt 1 X X ω q f αβ `wα [u(t)] `x αβ;q, wβ [u(t)] `x αβ;q. T α β q (10) Accuracy In equation (10), the x αβ;q are the quadrature points on A αβ and the ω q are the quadrature weights. If the expression under the integral on the right hand side of (8) is a polynomial of degree p in x and if the quadrature formula integrates exactly such polynomials, this step does not introduce any new discretization errors.

24 Cell-centered gradient reconstruction Interpolation theory: cell-centered gradient reconstruction on general grids A general theory of the gradient reconstruction is available (F. Haider). This theory is used in practice in CEDRE. Consistency of the slope Suppose given a function u(x) given by the (exact) averages u α on a given grid made of VF cells T α. Suppose given a linear slope reconstruction operator u σ α [u] = X β s αβ `uβ u α. (11) The first order consistency is equivalent to the tensor equation σ = X s αβ `hαβ σ for all σ R d. (12) β or equivalently X s αβ h αβ = I d d. (13) β

25 Cell-centered gradient reconstruction Interpolation theory: cell-centered gradient reconstruction on general grids A general theory of the gradient reconstruction is available (F. Haider). This theory is used in practice in CEDRE. Consistency of the slope Suppose given a function u(x) given by the (exact) averages u α on a given grid made of VF cells T α. Suppose given a linear slope reconstruction operator u σ α [u] = X β s αβ `uβ u α. (11) The first order consistency is equivalent to the tensor equation σ = X s αβ `hαβ σ for all σ R d. (12) β or equivalently X s αβ h αβ = I d d. (13) β

26 Least-square gradient reconstruction Proposition Let the matrix H α have rank d and let σ α R d be the solution of the least-squares problem 8 9 < X = min `uβ u α h σ R d : αβ σ 2 ;. (14) β W α Then σ α is unique and given by coefficients s αβ that are the columns of the minimum Frobenius norm solution to equation S αh α = I d d. (15)

27 Linear spectral analysis Asymptotic stability The stability property relevant for practical applications keeps to be the linear asymptotic stability. The semi-discrete MUSCL discrete form of the linear advection equation tu (x, t) + c u (x, t) = 0, (x, t) R d R +. (16) gives a dynamical system du α (t) dt = X β J αβ u β (t) ; 1 α N. (17)

28 Stability of the MUSCL scheme Matrix of the MUSCL scheme The operator J of the MUSCL scheme (17) is 8 < X J αβ = T α 1 (c n αγ) : + δ αβ + `c n αβ γ (18) + X γ X γ (n αγ c) + k αγ s αβ X (n αγ c) + k αγ s α δ αβ γ 9 (n γα c) + k γα s γβ + `n = βα c + k βα s β ;.

29 Stability of the MUSCL scheme, (cont.) Proposition (1) [ Stability of Linear Systems]The system (17) is stable in the sense that if and only if all eigenvalues λ of J satisfy 1 Re(λ) 0 where Re(λ) is the real part of λ. C = sup exp(tj) < (19) t 0 2 if Re(λ) = 0 then the Jordan index ı(λ) = 1 where ı(λ) is the maximal dimension of the Jordan blocks of J containing λ. Definition ( Stable finite-volume operator ) The spatial discretization operator J in (18) are called stable if all their eigenvalues satisfy properties (i)-(ii) of Proposition (1) for all advection velocities c R d.

30 Stability of the MUSCL scheme, (cont.) Proposition (1) [ Stability of Linear Systems]The system (17) is stable in the sense that if and only if all eigenvalues λ of J satisfy 1 Re(λ) 0 where Re(λ) is the real part of λ. C = sup exp(tj) < (19) t 0 2 if Re(λ) = 0 then the Jordan index ı(λ) = 1 where ı(λ) is the maximal dimension of the Jordan blocks of J containing λ. Definition ( Stable finite-volume operator ) The spatial discretization operator J in (18) are called stable if all their eigenvalues satisfy properties (i)-(ii) of Proposition (1) for all advection velocities c R d.

31 Localization of the eigenvalues of the MUSCL scheme Problem As usual in physics, we must solve the question of the localization of the eigenvalues of a linear problem. Here the matrix to analyze contains: The physics of the hyperbolic equation to solve The upwinding (numerical flux) The reconstruction operator The most difficult: the geometry of the grid. The shape of the grid influences the shape of the spectrum!

32 Localization of the eigenvalues of the MUSCL scheme (cont.) Classical tools Fourier analysis. Not possible on irregular grids. Direct algebraic eigenvalues analysis. Gerschgörin discs, field of values analysis: too weak. Energy methods, Lyapunov theorems.

33 Extended Lyapunov Theorem Theorem ( Extended Lyapunov Theorem) Let J M N (C). The following properties are equivalent 1 J satisfies the conditions of Proposition 0.2, that is all eigenvalues λ of J have Re(λ) 0 and if Re(λ) = 0 then the Jordan index ı(λ) = 1. 2 There exists a positive definite matrix G such that the matrix Q = GJ + J G is negative semidefinite.

34 Stability of the MUSCL scheme, (cont.) Corollaire Consider the initial value problem du(t) dt Then there is a constant C such that = Ju(t), u(0) = u 0, u(t) C N, J M N (C). (20) u(t) C u 0 for all t 0 if and only if there exists a positive definite matrix G such that the matrix Q = GJ + J G is negative semidefinite. Practical spectral analysis It turns out to be very difficult to find a matrix G that could give a rigorous proof of eigenvalue stability for the MUSCL operator J.

35 Stability of the MUSCL scheme, (cont.) Corollaire Consider the initial value problem du(t) dt Then there is a constant C such that = Ju(t), u(0) = u 0, u(t) C N, J M N (C). (20) u(t) C u 0 for all t 0 if and only if there exists a positive definite matrix G such that the matrix Q = GJ + J G is negative semidefinite. Practical spectral analysis It turns out to be very difficult to find a matrix G that could give a rigorous proof of eigenvalue stability for the MUSCL operator J.

36 Stability of the MUSCL scheme, (cont.) Table: Least-Squares Reconstruction on the First Neighborhood in 3D: spectral abscissa ω J and statistics of R α L(2, ) grid spectral abscissa average maximum 90th percentile tetrahedral tetrahedral e tetrahedral tetrahedral hybrid hybrid hybrid hybrid deformed cartesian e deformed cartesian e deformed cartesian e deformed cartesian e

37 Numerical experiment Figure: Tetrahedral grid. Unstable spectrum. Convection equation.

38 Numerical experiment Figure: Tetrahedral grid of a channel. The flow is along direction x.

39 Numerical experiment (cont.) log10(residuals) ρ ρ v x 4 6 ρ v y ρ v z ρ E time d Figure: Residuals history of [ρ, ρux, ρuy, ρuz, ρe], 800 time iterations, first dt neighborhood interpolation of the gradient, no slope limiters.

40 Numerical experiment (cont.) log10(residuals) ρ ρ v x ρ v y ρ v z ρ E time d Figure: Residuals history of [ρ, ρux, ρuy, ρuz, ρe], 4000 time iterations, dt second neighborhood interpolation of the gradient, no slope limiters.

41 High order finite volume MUSCL schemes Observation The MUSCL scheme with piecewise linear reconstruction is second order. This is sufficient for accurate computation of non linear waves such as shocks. This is dramatically insufficient for the computation of the fundamental characteritics of diffusive turbulent flows. The conclusion is that we need to have higher order spatial approximations. Question Is it possible to switch from a piecewise linear reconstruction to a higher order reconstruction within the MUSCL framework? Or should we give up the MUSCL approach (one unknown per cell) and switch to alternative high order schemes like the spectral element (Patera, Deville), the spectral difference (Wang,Liu,Vinokur) or the Discontinuous Galerkin schemes (Cockburn, Shu),... which all use more than one unknown per cell. The question to build higher order MUSCL schemes on irregular grids was already studied in the 90 (Harten, Osher, Abgrall,...). This keeps to be a question of fundamental practical importance!

42 High order finite volume MUSCL schemes Observation The MUSCL scheme with piecewise linear reconstruction is second order. This is sufficient for accurate computation of non linear waves such as shocks. This is dramatically insufficient for the computation of the fundamental characteritics of diffusive turbulent flows. The conclusion is that we need to have higher order spatial approximations. Question Is it possible to switch from a piecewise linear reconstruction to a higher order reconstruction within the MUSCL framework? Or should we give up the MUSCL approach (one unknown per cell) and switch to alternative high order schemes like the spectral element (Patera, Deville), the spectral difference (Wang,Liu,Vinokur) or the Discontinuous Galerkin schemes (Cockburn, Shu),... which all use more than one unknown per cell. The question to build higher order MUSCL schemes on irregular grids was already studied in the 90 (Harten, Osher, Abgrall,...). This keeps to be a question of fundamental practical importance!

43 High order MUSCL method Interpolation on irregular grids Basic methodology: interpolate the first, second and third order derivatives in a cell. The mathematical problem belongs therefore to the interpolation theory on unstructured grids. General theory of conservative interpolation on unstructured finite volume grids can be found in the thesis of F. Haider (2009, Univ. Paris 6). Fourth order MUSCL schemes The reconstructed function ũ i (x) is We have ũ i (x) = ū i + σ i (x x i ) θ i(x x i ) ψ i(x x i ) 3 (21) The scheme is as before: σ i u (x i ), θ i u (x i ), ψ i u (x i ) (22) U n+1 i = U n i t T i X A ij F(Ũi n, Ũn j ) (23) ij

44 High order MUSCL method Interpolation on irregular grids Basic methodology: interpolate the first, second and third order derivatives in a cell. The mathematical problem belongs therefore to the interpolation theory on unstructured grids. General theory of conservative interpolation on unstructured finite volume grids can be found in the thesis of F. Haider (2009, Univ. Paris 6). Fourth order MUSCL schemes The reconstructed function ũ i (x) is We have ũ i (x) = ū i + σ i (x x i ) θ i(x x i ) ψ i(x x i ) 3 (21) The scheme is as before: σ i u (x i ), θ i u (x i ), ψ i u (x i ) (22) U n+1 i = U n i t T i X A ij F(Ũi n, Ũn j ) (23) ij

45 Computation of the reconstruction Least square approximation The least-square slope is 8 >< >: σ α LS = P β V α c αβ (ū β ū α), (a) θα LS = P β V α c αβ (σ β σ α), (b) ψα LS = P β V α c αβ (θ β θ α), (c) (24) where the coefficients a α, b α and ā α depend only of the local shape of the grid. Approximate derivatives Suppose that σ α = u (x α), θ α = u (x α), ψ α = u (x α), Then approximations of σ α, θ α, ψ α of order 3, 2, 1 are given by the explicit formulas are given by 8 >< ψ α = P! β c αβ Pγ V β c βγ`σls γ σβ LS P δ V α c αδ (σδ LS σα LS ), (a) >: θ α = ã α ψα + P β c αβ(σβ LS σ α = ā α θα b α ψα + σ LS σ LS α ), (b) α, (c) (25)

46 Computation of the reconstruction Least square approximation The least-square slope is 8 >< >: σ α LS = P β V α c αβ (ū β ū α), (a) θα LS = P β V α c αβ (σ β σ α), (b) ψα LS = P β V α c αβ (θ β θ α), (c) (24) where the coefficients a α, b α and ā α depend only of the local shape of the grid. Approximate derivatives Suppose that σ α = u (x α), θ α = u (x α), ψ α = u (x α), Then approximations of σ α, θ α, ψ α of order 3, 2, 1 are given by the explicit formulas are given by 8 >< ψ α = P! β c αβ Pγ V β c βγ`σls γ σβ LS P δ V α c αδ (σδ LS σα LS ), (a) >: θ α = ã α ψα + P β c αβ(σβ LS σ α = ā α θα b α ψα + σ LS σ LS α ), (b) α, (c) (25)

47 Computational algorithm ALGORITHM 1 1 Compute σ LS α 2 Compute P β V α c αβ (σ LS β in all T α.this involves the stencil V α only. σα LS ) in all cells Tα. 3 Compute the third order derivative ψ α by (25)(a). 4 Compute the second order derivative θ α by (25)(b). 5 Compute the first order derivative σ α by (25)(c). Order of the interpolating polynomial The polynomial! p α(x) = ū α + σ α(x x α) + θ 1 α 2 (x xα) Tα ψ α(x x α) 3 (26) 6 is a fourth order reconstruction of u(x) in the cell T α.

48 Computational algorithm ALGORITHM 1 1 Compute σ LS α 2 Compute P β V α c αβ (σ LS β in all T α.this involves the stencil V α only. σα LS ) in all cells Tα. 3 Compute the third order derivative ψ α by (25)(a). 4 Compute the second order derivative θ α by (25)(b). 5 Compute the first order derivative σ α by (25)(c). Order of the interpolating polynomial The polynomial! p α(x) = ū α + σ α(x x α) + θ 1 α 2 (x xα) Tα ψ α(x x α) 3 (26) 6 is a fourth order reconstruction of u(x) in the cell T α.

49 Full discrete fourth order MUSCL scheme 4th order scheme Use the ordinary RK4 scheme. 8 k 0 = JU n k 1 = J(U n >< t k 0) k 2 = J(U n t k 1) k 3 = J(U n + t k 2 )! >: U n+1 = U n 1 + t 6 k k k k 3 (27) Observation: this scheme is stable on highly irregular random one-dimensional grid with random size. On a smooth varying grid the measured convergence rates are shown on the following table

50 Assessing the fourth order accuracy of the MUSCL scheme Rate of convergence of the 4th order MUSCL scheme - Sinusoidal grid. Period N=32 rate N=64 rate N=128 rate N=256 rate N= (-4) (-5) (-6) (-8) (-9) (-3) (-5) (-6) (-7) (-8) (-3) (-4) (-5) (-7) (-8)

51 The cubed sphere grid The cubed sphere grid Spherical grid made of 6 squares patches. The 6 patches are the projection of a cube on an (internal) sphere. Nonequiangular grid. Very interesting to discretize the SW system. First attempt recently done (Ulrich, Jablonowski, van Leer, JCP, 2010). Good results with some 4th order MUSCL scheme. Classical test-case (Williamson) are tested.

52 The cubed sphere grid (cont.) 1 N 0.5 E F B W 1 1 S N F E W 0 B S Figure: Cubed sphere grids, C8 and C32

53 Implementation so far CEDRE Second order MUSCL scheme in CEDRE (aerothermochemistry for internal flows). Various options of turbulence modling Various options of time-stepping Irregular general grids Multisolver/multidomain solver Fourth order accuracy Methodology of higher order accurate MUSCL scheme assessed on tetrahedral grids (F. Haider). Keeps the standard MUSCL framework,one unknown per cell. Extension to flows on the spherical earth Methodology seems possible on any kind of spherical grids: latitude/longitude, web grid, icosahedral grid, etc. A specifically interesting grid seems to be the cubed sphere grid.

54 Implementation so far CEDRE Second order MUSCL scheme in CEDRE (aerothermochemistry for internal flows). Various options of turbulence modling Various options of time-stepping Irregular general grids Multisolver/multidomain solver Fourth order accuracy Methodology of higher order accurate MUSCL scheme assessed on tetrahedral grids (F. Haider). Keeps the standard MUSCL framework,one unknown per cell. Extension to flows on the spherical earth Methodology seems possible on any kind of spherical grids: latitude/longitude, web grid, icosahedral grid, etc. A specifically interesting grid seems to be the cubed sphere grid.

55 Implementation so far CEDRE Second order MUSCL scheme in CEDRE (aerothermochemistry for internal flows). Various options of turbulence modling Various options of time-stepping Irregular general grids Multisolver/multidomain solver Fourth order accuracy Methodology of higher order accurate MUSCL scheme assessed on tetrahedral grids (F. Haider). Keeps the standard MUSCL framework,one unknown per cell. Extension to flows on the spherical earth Methodology seems possible on any kind of spherical grids: latitude/longitude, web grid, icosahedral grid, etc. A specifically interesting grid seems to be the cubed sphere grid.

56 In progress At ONERA Implementation of the fourth order MUSCL scheme for 3D gas dynamics problems (ONERA, F. Haider, B. Courbet). Cubed sphere grid High order Hermitian interpolation on the cubed sphere grid; Fast solver (FFT)

57 In progress At ONERA Implementation of the fourth order MUSCL scheme for 3D gas dynamics problems (ONERA, F. Haider, B. Courbet). Cubed sphere grid High order Hermitian interpolation on the cubed sphere grid; Fast solver (FFT)