Well-balanced and Asymptotic Schemes for Friedrichs systems

Transcription

1 B. Després (LJLL- University Paris VI) C. Buet (CEA) Well-balanced and Asymptotic Schemes for Friedrichs systems thanks to T. Leroy (PhD CEA), X. Valentin (PhD CEA), C. Enaux (CEA), P. Lafitte (Centrale Paris) and E. Franck (Inria) B. Després (LJLL-University Paris VI) C. Buet (CEA) thanks to T. Leroy (PhD CEA), X. Valentin (PhD CEA), C. Enaux (CEA), P. Lafitte (Centrale Paris) and E. Franck (Inria) Support of CEA is acknowledged p. 1 / 9

2 Section 1 Numhyp 015 p. / 9

3 Discretize tu + f (U) = S(U,... ) with FV-P 0 -cell centered schemes U n+1 j Tentative definition : U n j t + f n j+ 1 f n j 1 x = S(U n j,... ) WB-FV scheme = exact for stationary solutions AP (asymptotic preserving) schemes = WB with stiffness. Literature on WB : it is a one case after another strategy. Can we have a global view (classification) on WB methods? Numhyp 015 p. / 9

4 Some references Cargo-LeRoux, A WB for a model of an atmosphere with gravity CRAS I 318 (1994) Greenberg-Leroux, A WB scheme for the numerical processing of source terms in hyp. eq., Audusse-Bouchut-Bristeau-Klein-Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, (004). Bouchut. Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series, 004. Coquel-Godlewski. Asymptotic preserving scheme for Euler system with large friction, 011. T. Muller and A. Pfeiffer, Well-balanced simulation of geophysical flows via the shallow water equations with bottom topography : consistency and numerical computations, 014. Larsen-Morel-Miller, Asymptotic solution of num. transport solutions in optical thick regimes L. Gosse and G. Toscani, An AP WB scheme for the hyp. heat equations. CRAS I, 00. S. Jin, AP schemes for multiscale kinetic and hyperbolic equations : a review, 010. Buet-Després-Franck Design of AP schemes for the hyp. heat equation on unstructured meshes, 01. Buet-Després-Franck, AP Schemes on Distorted Meshes for Friedrichs Systems with Stiff Relaxation : Application to Angular Models in Linear Transport, 014. L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws..., 013. Numhyp 015 p. 3 / 9

5 Model problem : Friedrichs systems with relaxation A paradigm is the p-system with friction : linearization yields { tτ xu = 0, tu + xp(τ) = g σu. = { tτ 1 xu 1 = 0, tu 1 x ( c0(x) τ 1 ) = σu1. One gets the linear hyperbolic heat equation tu + x(a(x)u) = R(x)U, U = (c 0(x)τ 1, u 1) t, ( ) ( ) c where A = c 0(x) = A t and R = 0 (x) σ Notice the compatibility relation R + R t + A (x) 0 U t + (U, A(x)U) x = 1 ( R(x) + R(x) t + A (x)u, U ) 0. Numhyp 015 p. 4 / 9

6 S n and DOM Another example is DOM (Discrete Ordinates Method) for transfer Approximate I (x, t, µ) = ti + µ xi = σ (< I > I ). n n w i f i (x, t)δ(µ µ i ) + w i g i (x, t)δ(µ + µ i ). i=1 i=1 Normalize U = ( diag(w i ) f, diag(w i ) g) t R n, with w = ( w 1,..., w n, w 1,..., w n) R n. One gets the S n model tu + A xu = RU, A = A t, R = w w + I d, Numhyp 015 p. 5 / 9

7 Main idea Generic model problem is tu + x(a(x)u) + y (B(x)U) = R(x)U, U(t, x) R n, where x = (x, y) R, A(x) = A(x) t, B(x) = B(x) t. The vectorial space of stationary states, needed for WB analysis, is : U = {x U(x); x(a(x)u) + y (B(x)U) = R(x)U}. Main idea - consider the adjoint equation : tv + A(x) t xv + B(x) t y V = R(x) t V, V (t, x) R n. Magic property - this is just duality : t(u, V ) + x(a(x)u, V ) + y (B(x)U, V ) = 0. Numhyp 015 p. 6 / 9

8 Strategy The vectorial space of dual stationary states is V = { x V (x); A(x) t xv + B(x) t y V = R(x) t U }. Pick up V p V and define α p = (U, V p) R which is solution of a conservative equation tα p + x(a(x)u, V p) + y (B(x)U, V p) = 0 a) In 1D, assemble the system of conservation laws for α = (α p) R dim(v) = R dim(u). b) Discretize the new system of conservation laws. c) Write the new scheme for the original variable U. Numhyp 015 p. 7 / 9

9 For simplicity : A is constant with det(a) 0 So dim(v) = dim(u) = n. A t xv = R t V xv = A t R t V V (x) = e A t R t x V (0) where e A t R t x is a matrix exponential. Proposition : Define the change-of-basis matrix P(x) = e RA 1x, and the change of unknown α = P(x)U U = P 1 (x)α. The new conservative system rewrites tα + x(q(x)α) = 0 with a matrix defined through a global change of basis Q(x) = P(x)AP 1 (x) Note Q(x) is similar to A, so has the same eigenvalues. But the eigenvectors depend on x. Numhyp 015 p. 8 / 9

10 Discretization For additional simplification (once again), A is also symmetric : A = A t R n n. One has the spectral decomposition Au p = λ pu p, λ p 0 ( ) u p, U j+ 1 U j = 0, λ p > 0, ( ) u p, U j+ 1 U j+1 = 0, λ p < 0. Intermediate state L R The intermediate state is computed in function of L = j and R = j + 1. The FV discretization of tα + xβ = 0 with β = Q(x)α writes where β n j+ 1 α n+1 j α n j t + β n j+ 1 is the flux at time step t n = n t. β n j 1 x j = 0 Numhyp 015 p. 9 / 9

11 Diagonalisation of Q The right and left spectral decompositions of Q = Q(x ) = P(x )AP(x ) 1 Q are { Q(x )r p = λ pr p, r p = P(x )u p, Q t (x )s p = λ ps p, s p = P(x ) t u p. The solution at interface of the system t ( s p, β ) + λ p x ( s p, β ) = 0 yields a family of. One-state : defined by the well-posed linear system { ( s p, β β L ) = 0, λp > 0, ( s p, β β R ) = 0, λp < 0. It defines a first solver ϕ : R n R n R R n ϕ(β L, β R, x ) = β. Numhyp 015 p. 10 / 9

12 The WB scheme with the one-state solver First formulation with the global change of basis α n+1 α n j j t + ϕ(β n j,βn j+1,x j+ 1 ) ϕ(βj 1 n,βn j,x j 1 ) x j = 0 n ϕ(β 1 Second formulation Un+1 U n j j j + P(x t j ),βn j+1,x j+ 1 ) ϕ(βj 1 n,βn j,x j 1 ) x j = 0 Third purely local formulation, the one for implementation, U n+1 j U n j t U j+ + A 1 U j 1 x j + P(x j) 1 P(x j+ 1 ) I AU j+ x 1 + I P(x j) 1 P(x j 1 ) AU j j x 1 = 0. j where the flux U j+ 1 is solution of the linear system (with x ± j = x j± 1 x j ) ( ) u p, U j+ 1 e A 1 R x + j U j = 0, λ p > 0, ( ) u p, U j+ 1 e A 1 R x j+1u j+1 = 0, λ p < 0. Property : the scheme is WB. Proof : If β j = β for all j, the solution is stationary in the first formulation. Numhyp 015 p. 11 / 9

13 Schematic Structure of the one-state solver. Intermediate state N L L Two-states solver. Same principle but with upwinding of the eigenvectors { ( s L p, β β L ) = 0, λp > 0, ( s R p, β β R ) = 0, λp < 0, N R R It defines ψ : R n R n R R R n : ψ(β L, β R, x L, x R ) = β. Intermediate state N L N R L R Th. : If R + R t 0, then { } sp L { } s R λ p p>0 λ p<0 is linearly independent. Numhyp 015 p. 1 / 9

14 A 1 R = ( 0 σ 0 0 Interpretation for : t p + x u = 0, t u + x p = σu ) is nilpotent. So e A 1Rx = I + A 1 Rx. One-state solver = Steady state approximation of Huang-Liu 1986, or Jin-Levermore 1996 p j+ 1 u j+ 1 = p j + p j+1 = p j p j+1 + (1 σ x + j )u j (1 + σ x j+1 )u j+1, + (1 σ x + j )u j + (1 + σ x j+1 )u j+1. Two-state solver= Equal to the Gosse-Toscani scheme 00. p 1 σ x 1+σ x j+ 1 j+1 = (p +σ x + σ x j u j ) + + j (p +σ x + σ x j+1 u j+1 ), j j+1 j j+1 u 1 j+ 1 = (u j + u j+1 + p j p j+1 ). +σ x + σ x j j+1 - Immediate interest for non linear fluid with σ the friction parameter. Numhyp 015 p. 13 / 9

15 Section 3 Numhyp 015 p. 14 / 9

16 #1 : P n method, Valentin (PhD-CEA)+Enaux+Laffitte This is a challenging problem with a vast literature in neutron propagation. Machorro 007 : consider µ r u + 1 µ u = σu + q with r u(r, µ) = n i=0 u i(r)p i (µ) in the domain Solve tu + A r U + 1 GU = σu + b. r Valentin-Enaux-Lafitte propose to use Magnus series to compute the local matrix exponentials. Numhyp 015 p. 14 / 9

17 P n results : Machorro 007 Discontinuity of coefficients. flux dip, oscillations, higher cost with DG. Numhyp 015 p. 15 / 9

18 P n results : Valentin 015 No flux dip, no oscillations, reduced cost. Numhyp 015 p. 16 / 9

19 # : Transport-Doppler effect, Leroy (PhD-CEA)+Buet+D. Solve : tρ = κ ν νρ + σ(ν)(b(ν) ρ), v, t > 0 with κ = xu R. 3 The SWB (Spectrally Well Balanced) scheme writes where tρ j = ρ(ν ν, ρ ) = ρ e κ 3 ν ν σ(ν j ) ρ(ν j ν j+1, ρ j+1 ) ρ j 1 M(ν j+1, ν j ) ν j σ(s) s ds + 3 ν B(s)σ(s) s κ ν s τ is the stationary solution at ν j and the additional term M(ν j+1, ν) = e 3 κ ν j+1 ν j σ(s) s ds gives the correct limit κ 0. σ(τ) dτds τ Numhyp 015 p. 17 / 9

20 Upwing/WB/SWB WB and SWB correct WB non correct for κ = 0 Numhyp 015 p. 18 / 9

21 #3 : System with a zero eigenvalue This is a serious issue, related to resonance in hyperbolic theory. Consider P 1 model coupled a linear temperature equation tp + xu = τ(t p), tu + xp = σu, tt = τ(p T ), A = A t = and R = R t = τ 0 τ 0 σ 0 τ 0 τ Assume σ, τ > 0 : the solutions of the adjoint stationary equation satisfy T = p, so xû = 0 and x p = σû. Therefore. dim(v) = dim(u) = < 3 with basis 1 σx V 1 = 0 1 and V = 0 σx. Numhyp 015 p. 19 / 9

22 One abstract solution= shift the spectrum Consider progressive solutions of the adjoint equation { tv + A xv = RV, tv + ξ xv = 0, ξ R, where ξ R is arbitrary. It yields the shifted problem A ξ xv = RV, A ξ = A ξi. So det(a ξ ) 0. Probably non correct in the regime ξ 0, so useless. Numhyp 015 p. 0 / 9

23 Ad-hoc integration in time Set α 1 = (U, V 1) = p + T, α = (U, V ) = σx(p + T ) + u and α 3 = T tα 1 + x( σxα ( 1 + α ) = 0, ) tα + x (1 σ x )α 1 + σxα α 3 = 0, tα 3 = τ(α 1 α 3). The last equation is more an ODE (e τt α 3) = τe τt α 1 t α 3(x, t) = e τt τe τs α 1(x, s)ds + e τt α 3(x, 0). 0 Explicit Euler approximation over t reads α 3(x, t) 1 ( 1 e τ t) α 1(x, 0) + e τ t α 3(x, 0). It yields { tα 1 + x( σxα 1 + α ) = 0, tα + x ( (1 σ x )α 1 + σxα 1 ( 1 e τ t ) α 1 e τ t α 3(0) ) = 0. Numhyp 015 p. 1 / 9

24 Section 4 Numhyp 015 p. / 9

25 #4 : D Assume σ > 0 is constant tp + xu + y v = 0, tu + xp = σu, tv + y p = σv, Here det(a) = 0, det(b) = 0 and AB BA. The adjoint stationary states ( p, û, v) are xû + y v = 0, x p = σû, y p = σ v. So (û, v) = 1 p and p = 0. Therefore dim(v) = dim(u) = σ V = {( p, û, v); p is harmonic and (û, v) = 1σ } p. Numhyp 015 p. / 9

26 New formulation Set V n = V { p is an harmonic polynomial of degree n}. p V 1 is equivalent to p = a + bx + cy. V 1 yields three test functions 1 V 1 = 0 0, V = σx 1 0, V 3 = Set α i = (U, V i ) for i = 1,, 3. The new system is α 1 m 1 m t α + x α 1 + σxm 1 + y σxm α 3 σym σym σy 0 1 = 0 with m 1 = σxα 1 + α and m = σyα 1 + α 3. The original variable is recovered as U = P(x) 1 α where P(x) = σx 1 0 is non singular, and P(x)P(y) = P(y)P(x). σy 0 1 Numhyp 015 p. 3 / 9

27 Convergence Performed on the problem with an additional small parameter { tp + 1 ε u = 0, tu + 1 ε p = σ ε u, where σ > 0 is given and ε > 0 is a small parameter. In the limit ε 0 +, one gets asymptotically the parabolic equation tp 1 p = 0. σ Theorem (D on general grids) : The implicit edge-based-fv WB scheme (= Gosse-Toscani if 1D on uniform mesh) is convergent p ε h, t p ε, u ε h, t u ε L ([0,T ] Ω) C(h t 1 ), uniformly with respect to small parameter ε (0, 1] (numerics show convergence order between 1 and ). Sketch of the proof : linear estimates but of hyperbolic-parabolic type, balance between dissipativity and source terms, general grid, control of the constants,.... Numhyp 015 p. 4 / 9

28 Non AP feature of the standard edge based VF scheme element size L1 Norm L Norm h^ Error for the VF4 scheme Numhyp 015 p. 5 / 9

29 Meshes and P 1 (E. Franck) Initial data is Dirac mass at center Kershaw mesh Random mesh non WB-AP WB-FV edge, non AP WB-AP corner Mesh imprint vastly reduced with WB-AP corner-based-flux FV scheme. Numhyp 015 p. 6 / 9

30 Convergence order Analytical solution in the torus : p = f + ε tf and u = ε f with σ σ tf + ε σ t f 1 f = 0, σ f (t, x, y) = α(t) cos(lπx)) cos(lπy). α (t) + ε σ α (t) + L π σ α(t) = 0, α(t) = λ λ λ 1 e λ 1t λ 1 λ λ 1 e λ t : 1: 0.5: 0.5: 0: order one L Error e N In tests : ε = 0.01(40h) τ for τ {0, 1, 1, 1, }. 4 1 experimental convergence. Numhyp 015 p. 7 / 9

31 #5 : Lagrangian fluid dynamics, comparison standard FV versus WB-FV Initial stage of Rayleigh-Taylor instability : source term = gravity. D P1 standard FV spurious velocities WB-FV vortex is correct Numhyp 015 p. 8 / 9

32 Conclusions Unification of WB based on the dual equation essentially ideas : duality and linear, essentially families of WB. General principle : any WB-FV is a standard FV. AMC 015, Buet-D : and references therein. Validated for Friedrichs systems of large size inspired by radiation/neutron inspired problems. Any solver can be used : high order, even FEM is possible, or central schemes,... The change-of-basis matrix P(x) seems the good object to manipulate, particularly in multid. Ongoing : convergence estimates, zero eigenvalue problem. Open problems : - Comparison with other methods for transport equations. - Convergence for AP with this method : e A 1 R x ε - Extension to the full non linear case. Numhyp 015 p. 9 / 9