Semi-Conservative Schemes for Conservation Laws

Transcription

1 Semi-Conservative Schemes for Conservation Laws Rosa Maria Pidatella 1 Gabriella Puppo, 2 Giovanni Russo, 1 Pietro Santagati 1 1 Università degli Studi di Catania, Catania, Italy 2 Università dell Insubria, Como, Italy NumHyp 2015 Rosa Maria Pidatella, Gabriella Puppo, Giovanni Russo, Pietro Santagati

2 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

3 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

4 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

5 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

6 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

7 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

8 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

9 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

10 Outline Conservative form of the hyperbolic system Non Conservative form of the hyperbolic system Fully-Conservative FV Scheme Semi-Conservative FV Scheme Applications Classical GasDynamics Relativistic GasDynamics Results Conclusions

11 Hyperbolic System with Conservative Variables U t + f x (U) = 0, x Ω, t > 0 U(0, x) = U 0 (x), U(t, x) = U b (t) U R N conservative variables x Ω If U smooth U t + A(U)U x = 0, A(U) = f (U)

12 Hyperbolic Equations with Primitive Variables New set of variables V V = M(U), U = M 1 (V ) U t + A(U)U x = 0 V t + B(V )V x = 0 B = (M ) 1 AM

13 Shocks: Rankine-Hugoniot jump condition s = shock speed f (U l ) f (U r ) = s(u l U r )

14 Finite Volume Schemes Let s define the cell average values Ū j (t) = 1 x xj+1/2 x j 1/2 U(x, t) dx, j = 1,.., N Ūj n+1 = Ūj n t ( ) Fj+1/2 F j 1/2 x F j+1/2 = F (Ūn j p,..., Ūn j+q ) is the numerical flux function consistent with original system if F (Ūn,..., Ūn ) = f (Ū) Lipschitz continuity LWThrm guarantees that if U t U U weak solution

15 Fully Conservative methods dūj = 1 ( ) Fj+1/2 F j 1/2 dt x ( ) F j+1/2 = F U + j+1/2, U j+1/2 where the boundary extrapolated data U +, U obtained by a non oscillatory reconstruction from cell averages piecewise constant piecewise linear WENO

16 Fully Conservative methods Time discretization: explicit Runge-Kutta Ū (1) j = Ūn j Ū (i) j = Ūn j t x i 1 k=1 a ik ( F (k) j ) i = 2,..., ν F (i) j = F (U +(i) j+1/2, U (i) j+1/2 ) F (U+(i) j 1/2, U (i) ν Ū n+1 j = Ū n j t x i=1 b i ( F (i) j ) j 1/2 )

17 FV Semi-Conservative Scheme Ū n j rec U n j M(U n j ) V n j V (1) j Solve V t + B(V )V x = 0 with RK and find stage values V (i) j V (i) j = V (1) j t D x : derivative operator i 1 k=1 a ik B(V (k) j )D x V (k) j, i = 2,..., ν

18 Reconstruct V (i) j+1/2, V +(i) j+1/2 Compute F (U (i) j+1/2, U+(i) j+1/2 ) = F (V (i) j+1/2, V +(i) j+1/2 ) F (i) j and finally: = F (U (i) j+1/2, U+(i) j+1/2 ) F (U+(i) j 1/2, U (i) j 1/2 ) Ū n+1 j = Ū n j t x ν i=1 b i ( F (i) j )

19 Fully Conservative vs. Semi Conservative Schemes Fully Conservative based on ν RK stages requires ν reconstruction steps to get the boundary extrapolated data from each U (i) ; ν evaluations of the numerical fluxes F i based on U (i) ; 1 corrector step, with previously computed F i.

20 Fully Conservative vs. Semi Conservative Schemes Semi Conservative based on ν RK stages requires 1 reconstruction step, point values U n from Ūn 1 map evaluation V n = M(U n ) ν reconstruction steps to get the boundary extrapolated data from each V (i) and ν 1 steps to get the slope at the cell center x V (i) ; ν 1 matrix-vector A(V (i) ) x V (i) 2ν inverse map U (i)± = M 1 V (i)± ν evaluations of F i based on V (i) ; 1 corrector step, which assembles the fluxes F i.

21 Euler equations Conservative form ρ ρv E t + ρv ρv 2 + p (E + p)v x = 0

22 Conservative variables: Primitive variables: ρ, ρv, E = 1 2 ρv 2 + p γ 1 ρ, v and p

23 Euler equations Non conservative form ρ v p t + v ρ 0 0 v 1/ρ 0 γp v ρ v p x = 0.

24 Relativistic equations D t S t τ t + (Dv) x = 0 + (Sv+p) x = 0 + (S Dv) x = 0 Conservative variables: D = ρw, S = ρhw 2 v, τ = ρhw 2 p D Primitive variables: ρ, v and p

25 W = 1 1 v 2 v = S τ+p+d, W = (relativistic correction) τ+p+d (τ+p+d)2 S 2, ρ = D W, e = τ+p(1 W )+ρ(1 W 2 ) DW, p = ρe(γ 1) Then solve the non linear equation I(p; D, S, τ) = 0. to calculate ρ, v and p for a given set of D, S and τ. [Marti, Muller (1995)], [Ivanovski, PhDThesis (2010)]

26 Tests Sod test x [0, 1], γ = 1.4, t f = 0.2 (ρ L, v L, p L ) = (1.0, 0, 1.0) x < 0.5 (ρ R, v R, p R ) = (0.125, 0, 0.1) x > 0.5 Relativistic shock-tube problem x [0, 1], γ = 5/3, t f = 0.36 (ρ L, v L, p L ) = (10.0, 0, 13.3) x < 0.5 (ρ R, v R, p R ) = (1.0, 0, 0.0) x > 0.5

27 Fig.0 Accuracy of the schemes

28 Fig.1 Density for Euler eqs. with 100 points, Sod test, second order

29 Fig.2 Velocity for Euler eqs. with 100 points, Sod test, second order

30 Fig.3 Energy for Euler eqs. with 100 points, Sod test, second order

31 Fig.4 Density for Euler eqs. with 100 points, Sod test, fifth order

32 Fig.5 Velocity for Euler eqs. with 100 points, Sod test, fifth order

33 Fig.6 Energy for Euler eqs. with 100 points, Sod test, fifth order

34 Fig.7 Energy for Euler eqs. with 400 points, Sod test, fifth order

35 Fig.8 Enlargement for Energy, Euler eqs. with 400 points, Sod test, fifth order

36 Fig.9 Energy for Euler eqs. with 400 points, Sod test, fifth order, projection along characteristics

37 Fig.10 Density for Relativistic eqs., 400 points, Shock tube, second order

38 Fig.11 Velocity for Relativistic eqs., 400 points, Shock tube, second order

39 Fig.12 Pressure for Relativistic eqs., 400 points, Shock tube, second order

40 CPU: 1.4 GHz Elapsed Times (s) Order Euler FC/SC (400pts) / /4.7 5 C 137/116 Relativistic 2 nd order: 778/388

41 Work in progress High order scheme for relativistic eqs. Other Tests/Hyperbolic systems FD schemes Efficient implementation

42 Work in progress High order scheme for relativistic eqs. Other Tests/Hyperbolic systems FD schemes Efficient implementation

43 Work in progress High order scheme for relativistic eqs. Other Tests/Hyperbolic systems FD schemes Efficient implementation

44 Work in progress High order scheme for relativistic eqs. Other Tests/Hyperbolic systems FD schemes Efficient implementation

45 Conclusions More flexibility in the choice of unknown fields Efficiency comparable in Euler case but SC more efficient in the relativistic case

46 Conclusions More flexibility in the choice of unknown fields Efficiency comparable in Euler case but SC more efficient in the relativistic case

47 Thank you!

48 Numerical Flux ( F (U +, U ) = 1 2 f (U + ) + f (U ) ) α 2 (U+ U ), α coefficient of the Local LF numerical flux Re-write F in terms of primitive variables ) F (V +, V )= 2( f 1 (V + ) + f (V ) α 2 (M 1 (V + ) M 1 (V where f = f M 1 and f should be easier to compute than f.