Simplified ADER schemes based on a time-reconstruction GRP solver

Transcription

1 Simplified ADER schemes based on a time-reconstruction GRP solver Eleuterio F. TORO Professor Emeritus Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it Webpage: eleuteriotoro.com Joint work with Riccardo Dematté, Gino Montecinos, Christian Contarino and Vladimir Titarev Phil Roe s 80th Birthday Celebrations, 4th and 5th July Maxwell Centre, University of Cambridge 1 / 42

2 Happy Birthday PHIL from the CranFielD CFD days (...and nights) 2 / 42

3 Evolutionary PDEs We are interested in time-dependent partial differential equations: t Q(x, t) + A(Q(x, t)) = S(Q(x, t)) + D(Q(x, t)), x Ω, t > 0, ICs, BCs. (1) where Q(x, t) : vector of unknowns A(Q(x, t)) : advection operator in 1D, 2D or 3D D(Q(x, t)) : dissipative operator in 1D, 2D or 3D S(Q(x, t)) : source term, prescribed function of the unknown 3 / 42

4 ADER method in the finite volume framework t Q(x, t) + F(Q(x, t)) = S(Q(x, t)). (2) Exact integration of (2) in control volume [x i 1, x 2 i+ 1 ] [0, t] gives 2 where ˆQ n+1 i = ˆQ n i t x (ˆF i+ 1 2 ˆQ n i = 1 xi+ 1 2 Q(x, t n )dx, x x i 1 2 ˆF i 1 ) + tŝi, (3) 2 ˆF i+ 1 2 = 1 t t n+1 F(Q(x i+ 1, t))dt, t n 2 (4) Ŝ i = 1 t n+1 xi+ 1 2 S(Q(x, t))dxdt. t x t n x i / 42

5 ADER finite volume method in 1D One-step, fully discrete scheme. Numerical flux Q n+1 i Numerical source = Q n i t x (F i+ 1 2 F i+ 1 2 S i 1 t x 1 t t 0 t xi F i 1 ) + ts i. (5) 2 F(Q i+ 1 (τ))dτ (6) 2 x i 1 2 S(Q i (x, τ))dxdτ (7) t n+1 t Q n+1 i F i 1 2 S i F i+ 1 2 t n Q n i x i 1 2 x i x i+ 1 2 x 5 / 42

6 Spatial reconstruction, eg. polynomials Any reconstruction on fixed stencils (linear) TVD TVB ENO WENO 6 / 42

7 The Generalized Riemann Problem (GRP) Generalized Riemann problem for hyperbolic systems of balance laws: PDEs: t Q + x F(Q) { = S(Q), x (, ), t > 0 QL (x) if x < 0 ICs: Q(x, 0) = (8) Q R (x) if x > 0 Related works: Glimm et al. (1984) Ben-Artzi and Falcovitz (1984) Li Tatsien (1985) Harten et al. (1987) LeFloch and Raviart (1989) Men Shov (1990) LeFloch and Tatsien (1991) Toro and Titarev (2002) 7 / 42

8 Conventional and Generalized Riemann Problems q L q(x, 0) q L (x) q(x, 0) q R q R (x) x = 0 t x x = 0 t x 0 x 0 x Local initial conditions and structure of solution of Riemann problem: Left side: classical case; Right side: generalized case. 8 / 42

9 Generalized Riemann Problem: example Example: Structure of the solution of the Generalized Riemann problem for the Euler equations (Courtesy: Dr V A Titarev). 9 / 42

10 ADER (*) (**) has two main ingredients: High-order non-linear spatial reconstruction (once per time step); Generalized Riemann problem, GRP Initial conditions are piece-wise smooth (e.g. polynomials of any degree), and Source terms are included In this talk we introduce a new high-order non-linear time reconstruction (once per time step) as a way to solve the GRP (*) EF Toro, RC Millington and LAM Nejad. Towards Very High-Order Godunov Schemes. In Godunov Methods: Theory and Applications. Edited Review. E. F. Toro (Editor). Pages , Kluwer Academic/Plenum Publishers, (**) T Schartzkopf, CD Munz and EF Toro. ADER: High-order approach for lindar hyperbolic systems in 2D. J Scientific Computing. Vol. 17, pp , / 42

11 A solver for the generalized Riemann problem (*) We seek an approximate solution Q LR (τ) at x = 0 as as function of time Q LR (τ) = Q(0, 0 + ) + The leading term of the expansion K k=1 [ ] (k) τ k t Q(0, 0 + ) k! (9) Q(0, 0 + ) = lim t 0 + Q(0, t) (10) is computed by solving conventional homogeneous Riemann problem. t 0 x Seek solution at interface (along t-axis) as function of time (*) Toro E F and Titarev V A. Solution of the generalized Riemann problem for advection-reaction equations. Proceedings of the Royal Society of London A, Vol. 458, pp , / 42

12 Key aspects of the solver Cauchy-Kovalevskaya procedure: (k) t Q(x, t) = P (k) ( (0) x Q, x (1) Evolution equations for spatial derivatives: Q,..., (k) x Q) (11) t ( x (k) Q(x, t)) + A(Q) x ( x (k) Q(x, t)) = H (k) (12) Riemann problems for spatial derivatives: PDEs: ICs: Complete solution: t ( (k) x Q(x, t)) + A (0) LR x( (k) x Q(x, t)) = 0 x (k) Q(x, 0) = Q (k) L (0) if x < 0 Q (k) R (0) if x > 0 (13) Q LR (τ) = Q(0, 0 + ) + K k=1 [ ] (k) τ k t Q(0, 0 + ) k! (14) 12 / 42

13 Example: The Baer-Nunziato equations Fluid 1 pressure Reference solution (MH) DRP 0 DRP 1 DRP 2 DRP Time (s) Baer-Nunziato equations. GRP solution for solid phase pressure (Castro CE and Toro EF, Journal of Computational Physics 227 (2008) , 2008) 13 / 42

14 Solvers for the GRP used in ADER, so far... 1 EF Toro and VA Titarev. Solution of the generalized Riemann problem for advection-reaction equations. Proceedings of the Royal Society of London. Series A. Vol. 458, pages , CE Castro and EF Toro. Solvers for the high-order Riemann problem for hyperbolic balance laws. Journal of Computational Physics. Vol. 227, pages , M Dumbser, C Enaux and EF Toro. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Journal of Computational Physics. Vol. 227, pages , CR Götz and A Iske. Approximate Solutions of Generalized Riemann Problems for Nonlinear Systems of Hyperbolic Conservation Laws. Mathematics of Computation 85, pp 35-62, EF Toro and GI Montecinos. Implicit, semi-analytical solution of the generalized Riemann problem for stiff hyperbolic balance laws. Journal of Computational Physics. Volume 303, Pages , CR Götz and M Dumbser. A novel solver for the generalized Riemann problem based on a simplified LeFlochRaviart expansion and a local space-time discontinuous Galerkin formulation, J. Scientific Computing, Vol. 69, Issue 2, pp , EF Toro, R Dematté, GI Montecinos, C Contarino and VA Titarev. Simplified ADER schemes based on a time-reconstruction GRP solver. IN PREPARATION. 14 / 42

15 Acknowledgements on ADER work Mauricio Cáceres, Neal Cheney, Richard Millington,Vladimir Titarev, Thomas Schwarzkopff,Claus-Dieter Munz, Michael Dumbser, Yoko Takakura,Martin Kaeser, Armin Iske, Cedric Enaux, Cristobal Castro, Giovanni Russo, Carlos Parés, Dinshaw Balsara, Manuel Castro, Arturo Hidalgo, Gianluca Vignoli, Giovanna Grosso, Matteo Antuono, Alberto Canestrelli, Annunziato Siviglia, Gino Montecinos, Lucas Müller, Junbo Cheng, Jiang Song, Claus Götz, Christian Contarino, Olindo Zanotti, Raphaël Loubére, Steven Diot, Marina Strochi, Morena Celant, Riccardo Dematté / 42

16 New solver for the generalized Riemann problem 16 / 42

17 Idea: time-reconstruction to compute time derivatives We start from Q LR (τ) = Q(0, 0 + ) + K k=1 [ ] (k) τ k t Q(0, 0 + ) k! (15) where The leading term as before For time-derivatives, introduce time-reconstruction polynomials Cauchy-Kovalevskaya procedure avoided, for high-order derivatives Several possible methods are possible Here we discuss two 17 / 42

18 Methods 18 / 42

19 Time-reconstruction polynomial V(t) at the interface V(t n k + ) are interpolant points of V(t) L = K for TR0 and L = K 1 for TR1 t QLR(τ) t n+1 t n V (t n +) t n x t n 1 V (t n 1 + ) t n 1 t n l+1 t n l V (t n l + ) t n l t n L+1 t n L V (t n L + ) t n L x i / 42

20 Method 1: TR0 Q LR (τ) = Q(0, 0 + ) + K k=1 [ ] (k) τ k t Q(0, 0 + ) k! (16) Leading term of expansion as before Time derivatives computed from reconstructed function V(t) Interpolant points V(t n k + ) for V(t) are: V(t n k + Time derivatives are ) = Q(0, tn k + ), k = 0,..., K (17) (k) t Q(0, 0 + ) = dk dt k V(tn +), k = 1,..., K (18) 20 / 42

21 Method 2: TR1 Q LR (τ) = Q(0, 0 + ) + (1) t Q(0, 0 + ) + Leading term of expansion as before First time derivative as K k=2 [ ] (k) τ k t Q(0, 0 + ) k! (19) (1) t Q(0, 0 + ) = A x Q + S(Q) (20) x Q computed as in Toro-Titarev solver Higher-order time derivatives computed from reconstructed function V(t) Interpolant points V(t n k + ) for V(t) are: V(t n k + ) = ( A xq + S(Q)) (0,t n k ), k = 0,..., K 1 (21) Time derivatives are + (k) t Q(0, 0 + ) = dk 1 dt k 1 V(tn +), k = 2,..., K 1 (22) 21 / 42

22 Stability, linear advection reaction equation (LARE) 22 / 42

23 Empirical stability study 23 / 42

24 Linear and non-linear TR1 schemes 24 / 42

25 Convergence rates 25 / 42

26 ADER 5 and ADER 7 for LARE t q(x, t) + λ x q(x, t) = βq(x, t) (23) Mesh L 1 -err L 1 -ord L 2 -err L 2 -ord L -err L -ord e-04 [-] 5.98e-04 [-] 1.46e-03 [-] e e e e e e e e e e e e ADER 5 TR1-W for linear-advection-reaction equation. CF L = 0.4; T out = 1s; λ = 1; β = 0.2 Mesh L 1 -err L 1 -ord L 2 -err L 2 -ord L -err L -ord e-05 [-] 6.21e-05 [-] 9.33e-05 [-] e e e e e e e e e ADER 7 TR1-W for linear-advection-reaction equation. CF L = 0.2; T out = 1s; λ = 1; β = / 42

27 ADER 6 and ADER 8 for LARE t q(x, t) + λ x q(x, t) = βq(x, t) (24) Mesh L 1 -err L 1 -ord L 2 -err L 2 -ord L -err L -ord e-07 [-] 8.24e-07 [-] 1.37e-06 [-] e e e e e e e e e ADER 6 TR1 for linear-advection-reaction equation. CF L = 0.3; T out = 1s; λ = 1; β = 0.2 Mesh L 1 -err L 1 -ord L 2 -err L 2 -ord L -err L -ord e-06 [-] 7.99e-06 [-] 1.28e-05 [-] e e e e e e ADER 8 TR1 for linear-advection-reaction equation. CF L = 0.15; T out = 1s; λ = 1; β = / 42

28 ADER 5 for Euler equations with exact solution Mesh L 1 -err L 1 -ord L 2 -err L 2 -ord L -err L -ord e-04 [-] 8.22e-04 [-] 1.71e-03 [-] e e e e e e e e e ADER 5 with time-reconstruction GRP solver (TR1-W, present method): CF L = 0.4; T out = 2s. Mesh L 1 -err L 1 -ord L 2 -err L 2 -ord L -err L -ord e-04 [-] 9.10e-04 [-] 1.98e-03 [-] e e e e e e e e e ADER 5 with Toro-Titarev GRP solver: CF L = 0.9; T out = 2s. 28 / 42

29 Numerical Results 29 / 42

30 Long time evolution, Euler equations 30 / 42

31 Riemann problems and shocks 31 / 42

32 Test 4 32 / 42

33 Test 6 33 / 42

34 Shock-turbulence interaction, Euler equations 34 / 42

35 Efficiency, Euler equations 35 / 42

36 3rd order / 42

37 4th order / 42

38 5th order / 42

39 6th order / 42

40 7th order / 42

41 Concluding Remarks Very high order methods are essential for these reasons: Growing trend to use PDEs to understand the physics they embody Only very accurate solutions of PDEs will achieve this and also reveal (i) limitations of mathematical models (PDEs themselves) (ii) uncertainty on parameters of the problem Key issue: isolate errors Efficiency is the thing: given a error deemed acceptable, then high order methods attain that error, if small, much more efficiently, in fact by orders of magnitude Very long time evolution simulations: high order methods are essential In this talk: Simplified version of the ADER methods to ease implementation and application Results are preliminary but encouraging Many pending issues, eg stiff source terms and assesment in 2D, 3D. 41 / 42

42 THANK YOU 42 / 42