DDFV Schemes for the Euler Equations

Transcription

1 DDFV Schemes for the Euler Equations Christophe Berthon, Yves Coudière, Vivien Desveaux Journées du GDR Calcul, 5 juillet 2011

2 Introduction Hyperbolic system of conservation laws in 2D t W + x f(w)+ y g(w) = 0 W : R 2 R + Ω R d : unknown state vector f,g : Ω R d : flux functions Ω convex set of physical states Objective : derive a numerical scheme Second order accurate Ω preserving Unstructured meshes CFL restriction Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 2 / 24

3 Example : the 2D Euler equations W = ρ ρu ρv E, f(w) = ρu ρu 2 + p ρuv u(e + p), g(w) = ρv ρuv ρv 2 + p v(e + p) where ρ is the density, (u,v) the velocity, E the total energy and p the pressure given by the perfect gas law ( p = (γ 1) E ρ ( u 2 + v 2)) 2 Set of physical states { Ω = W R 4 ;ρ > 0,(u,v) R 2,E ρ ( u 2 + v 2) > 0} 2 Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 3 / 24

4 1 MUSCL schemes 2 Robustness and CFL restriction 3 Reconstruction Procedure: DDFV method 4 Numerical results Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 4 / 24

5 Meshes notations K j2 n ij2 T ij2 l ij1 K j3 K j1 T ij3 T ij1 n ij1 n ij3 l i j1j5 θ ij1 n i j1j5 T ij4 T ij5 The primal mesh and the dual mesh K j5 K j4 n ij4 n ij5 A primal or dual cell K i Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 5 / 24

6 First-order scheme W n+1 i = W n i t K i l ij φ ( W n i,w n j,θ ij ) φ 1D Godunov-type flux : under the CFL restriction t x max{λ± (W L,W R,θ)} 1, we have 2 φ(w L,W R,θ) = h θ (W L )+ x 2 t W L 1 t 0 x 2 W θ ( x t,w L,W R )dx hθ = cosθf + sinθg : flux in the direction θ Wθ approximate Riemann solver valued in Ω. Consistency : φ(w,w,θ) = h θ (W) Conservation : φ(w L,W R,θ) = φ(w R,W L,θ+π) Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 6 / 24

7 MUSCL scheme First-order scheme on the cell K i W n+1 i = W n i t K i l ij φ ( W n i,w n j,θ ij ), Second-order MUSCL scheme on the cell K i W n+1 i = W n i t K i l ij φ(w ij,w ji,θ ij ), K i K j W ij W ji W j W ij and W ji second-order approximations at the interface between K i and K j. How to compute W ij? W i Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 7 / 24

8 1 MUSCL schemes 2 Robustness and CFL restriction 3 Reconstruction Procedure: DDFV method 4 Numerical results Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 8 / 24

9 Motivations Usual CFL restriction on a square (first-order): t l max {λ± (Wi n,wj n,θ ij )} 1 4 Usual CFL restriction on a quadrilateral (first-order): t K i max{ l ij λ ± (Wi n,wj n,θ ij )} 1 8 Inconsistency. The CFL restriction for quadrilateral is not optimal. Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 9 / 24

10 First-order scheme: CFL restriction Under the CFL restriction t x max {λ± (Wi n,wj n,θ ij )} 1, we have 2 W n+1 i = 1 x 2 K i l ij Wi n t K i + 1 K i l ij 0 x 2 l ij h θij (W n i ) ( x W θij t,wn i,wj n ) dx Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 10 / 24

11 First-order scheme: CFL restriction Under the CFL restriction t x max {λ± (Wi n,wj n,θ ij )} 1, we have 2 W n+1 i = 1 x 2 K i ( l ij h θij (Wi n f ) = g l ij Wi n t K i + 1 K i ) (W n i ) l ij 0 x 2 l ij h θij (W n i ) ( x W θij t,wn i,wj n ) dx l ij n ij = 0 by Green s formula Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 10 / 24

12 First-order scheme: CFL restriction Under the CFL restriction t x max {λ± (Wi n,wj n,θ ij )} 1, we have 2 W n+1 i = 1 x 2 K i We define P i = l ij Wi n K i 0 l ij x 2 l ij and we take x = 2 K i P i 1 x 2 K i l ij = 0 ( x ) W θij t,wn i,wj n dx Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 10 / 24

13 The CFL restriction becomes and we have with W n+1 ij W n+1 i = 1 K i t K i P { i max λ ± (Wi n,wj n,θ ij ) } 1 = 1 P i 0 ( x ) l ij Wθij K i t,wn i,wj n dx P i l ij W n+1 ij = P 0 ( i x ) Wθij K i K i t,wn i,wj n dx P i W n+1 ij Ω as the mean value of a function valued in the convex Ω. W n+1 i Ω as a convex combination of the W n+1 ij. Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 11 / 24

14 Theorem : Robustness of the first-order scheme If the following hypothesis are satisfied (i) W n i Ω, i Z (ii) We have the CFL condition Then the states W n+1 i remain in Ω. Remark : this CFL can be written t P i K i max { λ ± (Wi n,wj n,θ ij ) } 1, i t l i K i max { λ ± (Wi n,wj n,θ ij ) } 1 p i p i number of edges of the cell K i l i = 1 p i P i mean length of the edges Consistency with the CFL restriction for a square Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 12 / 24

15 Second-order MUSCL scheme We define the intermediate states Wij n+1 = W ij t l ij φ(w ij,w ji,θ ij )+ T ij k ν(i,j) l i jk φ(w ij,w ik,θjk i ) which are the updated states by the first-order scheme on the subcell T ij. By the previous theorem, if the CFL restriction t P ij { max λ ± (W ij,w ji,θ ij ),λ ± (W ij,w ik,θjk i T ij )} 1 k ν(i,j) is satisfied, then W n+1 ij is in Ω. 1 K i T ij W n+1 ij = T ij K i W ij t K i This state is in Ω as a convex combination of states in Ω. φ(w ij,w ji,θ ij ) Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 13 / 24

16 Theorem : Robustness of the MUSCL scheme If the following hypothesis are satisfied (i) The initial states W n i are in Ω (ii) The reconstructed states W ij are in Ω (iii) The reconstruction satisfies the conservation property T ij K i W ij = Wi n (iv) We have the CFL condition i Z t max P ij T ij Then the states W n+1 i remain in Ω. { max λ ± (W ij,w ji,θ ij ),λ ± (W ij,w ik,θjk i )} 1 k ν(i,j) Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 14 / 24

17 1 MUSCL schemes 2 Robustness and CFL restriction 3 Reconstruction Procedure: DDFV method 4 Numerical results Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 15 / 24

18 Computation of the states W ij S3 Q2 S2 W3 Ŵ2 W2 T2 Q1 Ŵ1 Q3 T3 T1 Ŵ3 S0 W0 S1 W1 T4 T5 S4 W4 Q5 Ŵ5 Q4 Ŵ4 S5 W5 Geometry of the cell K The reconstructed states have to satisfy: Known states and reconstructed states Ŵ j Ω T ij K i Ŵj = W 0 If we take Ŵj = W(Q j ) with W a linear function on K, we have T ij K i Ŵj = W 0 W(S 0 ) = W 0 Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 16 / 24

19 1 Gradient reconstruction (DDFV) We define a continuous function W : K R d piecewise linear on each triangle T j and such that W(S j ) = W j, j ν(i). 4 3 T3 T4 0 T2 T1 T Projection For 1 k d, we define E k (ν) = W k (X) [(W 0 ) k +ν (X S 0 )] 2 dx, K where the subscript k denotes the k-th component. Let µ R d be the vector whose k-th component is the solution of E k (µ k ) = min ν R 2 E k(ν). We define W µ (X) : K R d the linear function whose k-th component is (W 0 ) k +µ k (X S 0 ). Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 17 / 24

20 3 Limitation of the slope µ We restrict Ω to a close set Ω ǫ. In the Euler case, { Ω ǫ = W R 4 ;ρ ǫ,(u,v) R 2,E ρ ( u 2 + v 2) ǫ}. 2 We define the optimal slope limiter by { } α = max t [0,1], W tµ (Q j ) Ω ǫ, j ν(i). 4 Finally, the reconstructed states are given by Ŵj = W αµ (Q j ). Limitation procedure Ŵj Ω W(S 0 ) = W 0 T ij K i Ŵj = W 0 The DDFV-MUSCL scheme is robust Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 18 / 24

21 Computation of the states at the vertices We write a MUSCL scheme on both the primal and the dual mesh The states at a vertex of the dual mesh is exactly the state at the center of the associated primal cell We approximate the state at a vertex of the primal mesh by the state at the center of the associated dual cell Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 19 / 24

22 1 MUSCL schemes 2 Robustness and CFL restriction 3 Reconstruction Procedure: DDFV method 4 Numerical results Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 20 / 24

23 Meshes Square mesh Quadrilateral mesh Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 21 / 24

24 Four shocks Square mesh DDFV-MUSCL scheme Square mesh MUSCL scheme Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 22 / 24

25 Four shocks Square mesh DDFV-MUSCL scheme Quadrilateral mesh DDFV-MUSCL scheme Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 22 / 24

26 Four contact discontinuities Square mesh DDFV-MUSCL scheme Square mesh MUSCL scheme Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 23 / 24

27 Four contact discontinuities Square mesh DDFV-MUSCL scheme Quadrilateral mesh DDFV-MUSCL scheme Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 23 / 24

28 Perspectives Allow non-conservative reconstructions, i.e. which don t satisfy T ij K i W ij = Wi n Optimization of the CFL condition in the robustness theorem for the MUSCL scheme Better approximation of the value at the vertices of the primal mesh, especially in the case of very distorded meshes Distorded mesh Vivien Desveaux DDFV Schemes for the Euler Equations Journées du GDR Calcul 24 / 24