All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach coefficients

Transcription

1 All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach coefficients Christophe Chalons LMV, Université de Versailles Saint-Quentin-en-Yvelines Joint works with M. Girardin and S. Kokh 1/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

2 Outline Introduction 1 Introduction /59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

3 Outline Introduction 1 Introduction /59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

4 Introduction Motivation : numerical study of two-phase flows in nuclear reactors We consider the following model t ρ + (ρu) = 0 t (ρu) + (ρu u) + p = 0 t (ρe) + [(ρe + p)u] = 0 where ρ, u = (u, v) t, E denote respectively the density, the velocity vector and the total energy of the fluid. Let e = E u 2 2 be the specific and τ = 1/ρ the covolume 4/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

5 Introduction We are especially interested in the design of numerical schemes when the model depends on a parameter ɛ > 0. The following three flow regimes are of interest Classical regime : ɛ = O(1) Low ɛ regime : ɛ << 1 Limit regime : ɛ 0 5/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

6 Outline Introduction 1 Introduction /59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

7 Large friction regime We consider the following model with friction and gravity t ρ + (ρu) = 0 t (ρu) + (ρu u) + p = ρ(g αu) t (ρe) + [(ρe + p)u] = ρu.(g αu) where g, α denote the gravity field and the friction coefficient. The large friction regime is obtained by replacing α with α ɛ t ρ + (ρu) = 0 t (ρu) + (ρu u) + p = ρ(g α ɛ u) t (ρe) + [(ρe + p)u] = ρu.(g α ɛ u) with ɛ << 1 7/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

8 Large friction regime Remark 1. Setting u = u 0 + ɛu 1 + O(ɛ 2 ), the long time behaviour of the solutions is given by u 0 = 0 t ρ + (ρu 1 ) = 0 p = ρ(g αu 1 ) t (ρe) + [(ρe + p)u 1 ] = ρu 1.(g αu 1 ) see Hsiao-Liu, Nishihara, Junca-Rascle, Lin-Coulombel, Coulombel-Goudon, Marcati-Milani... for rigorous proofs Remark 2. This system will not be considered in the design of the numerical strategy 8/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

9 Low Mach regime Introduction Introducing the characteristic and non-dimensional quantities : x = x L, t = t T, ρ = ρ ρ 0, u = u u 0, v = v v 0, e = e e 0, p = p p 0, c = c c 0 with u 0 = v 0 = L T, e 0 = p 0 ρ 0 and p 0 = ρ 0 c0 2, the non-dimensional system is (no gravity, no friction) t ρ + (ρu) = 0 t (ρu) + (ρu u) + 1 M 2 p = 0 t (ρe) + [(ρe + p)u] + M2 2 ( t(ρu.u) + (ρu.uu)) = 0 where M = u 0 c 0 denotes the Mach number and plays the role of ɛ 9/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

10 Low Mach regime Introduction t ρ + (ρu) = 0 t (ρu) + (ρu u) + 1 M 2 p = 0 t (ρe) + [(ρe + p)u] + M2 2 ( t(ρu.u) + (ρu.uu)) = 0 Definition. The flow is said to be in the low Mach regime if M << 1 and p = O(M 2 ) Remark 1. Using asymptotic expansions in powers of M in the governing equations of ρ, u, p and boundary conditions leads to t ρ 0 + (ρ 0 u 0 ) = 0 t u 0 + (u 0 )u ρ 0 p 2 = 0 u 0 = 0 Remark 2. This system will not be considered in the design of the numerical strategy 10/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

11 Numerical issue in the Low Mach regime Accurate time-explicit computations of solutions generally require a mesh size h = o(m) a time step t = O(hM) which is out of reach in practice More details can be found in the large body of literature on this subect : A. Mada, E. Turkel, H. Guillard, C. Viozat, B. Thornber, S. Dellacherie, P. Omnes, P-A. Raviart, F. Rieper, Y. Penel, P. Degond, S. Jin, J.-G. Liu, P. Colella, K. Pao, E. Turkel, R. Klein, J-P Vila, M.G., B. Després, M. Ndinga, J. Jung, M. Sun,... General cure : change the treatment of acoustic waves in the low Mach regime by centering the pressure gradient 11/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

12 Numerical issue in the large friction regime Accurate time-explicit computations of solutions generally require a mesh size h = o(ɛ) a time step t = O(ɛ) which is out of reach in practice More details can be found in the large body of literature on this subect : L. Hsiao, T.-P. Liu, S. Jin, L. Pareschi, L. Gosse, G. Toscani, F. Bouchut, H. Ounaissa, B. Perthame, C. C., F. Coquel, E. Godlewski, P.-A. Raviart, N. Seguin, C. Berthon, P.-G. LeFloch, R. Turpault, F. Filbet, A. Rambaud, M. Girardin, S. Kokh, C. Cancès, H. Mathis, N. Seguin, S. Cordier, B. Després, E. Franck, C. Buet,... General cure : upwinding of the source terms at interfaces (USI) 12/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

13 A couple of definitions Uniform stability A scheme is said to be stable in the uniform sense if the CFL condition is uniform with respect to ɛ This avoids stringent CFL restrictions t = O(hM) or t = O(ɛ) Uniform consistency A scheme is said to be consistent in the uniform sense if the truncation error is uniform with respect to ɛ This avoids large numerical diffusion and mesh size restrictions h = o(m) or h = O(ɛ) All-regime scheme A scheme is said to be all-regime if it is able to compute accurate solutions with a mesh size h and a time step t independent of ɛ 13/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

14 Obectives Introduction Our obective is to propose a numerical scheme that is all-regime : uniform stability and uniform consistency w.r.t. ɛ able to deal with any equation of state multi-dimensional on (possibly) unstructured meshes How to do that... 14/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

15 Outline Introduction 1 Introduction /59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

16 How to reach these obectives How to get the uniform stability? - implicit treatment of the fast phenomenon - explicit treatment of the slow phenomenon (sake of accuracy) Lagrange-Proection strategy Coquel-Nguyen-Postel-Tran How to get the uniform consistency? - modify the numerical fluxes to reduce the numerical diffusion Truncation errors in equivalent equations How to deal with any (possibly strongly nonlinear) pressure law p? - overcome the non linearities, linearization Relaxation strategy Suliciu, Jin-Xin, Bouchut, C.-Coquel, C.-Coulombel How to deal with unstructured meshes in multi-d? - work on a fixed mesh (no need to deform unstructured meshes) Operator splitting strategy and rotational invariance 16/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

17 Lagrange-Proection strategy Let us first focus on the 1D system t ϱ + x ϱu = 0 t ϱu + x (ϱu 2 + p) = 0 t (ϱe) + x (ϱeu + pu) = 0 Using chain rule arguments, we also have t ϱ + u x ϱ + ϱ x u = 0 t ϱu + u x ϱu + ϱu x u + x p = 0 t ϱe + u x ϱe + ϱe x u + x pu = 0 so that splitting the transport part leads to t ϱ + ϱ x u = 0 t ϱu + ϱu x u + x p = 0 t ϱe + ϱe x u + x pu = 0 Lagrangian-step t ϱ + u x ϱ = 0 t ϱu + u x ϱu = 0 t ϱe + u x ϱe = 0 Transport-step 17/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

18 Lagrange-Proection strategy The Lagrangian-step t ϱ + ϱ x u = 0 t ϱu + ϱu x u + x p = 0 t ϱe + ϱe x u + x pu = 0 with τ = 1/ϱ and τ x = m. also writes t τ m u = 0 t u + m p = 0 t E + m pu = 0 Eigenvalues are given by ρc, 0, ρc Usual CFL conditions for time-explicit schemes write t h max(ρc) 1 2 The idea is to propose a time-implicit scheme to avoid this time-step restriction ( t = O(hM) in the low Mach regime) 18/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

19 Lagrange-Proection strategy The Transport-step is t ϱ + u x ϱ = 0 t ϱu + u x ϱu = 0 t ϱe + u x ϱe = 0 also writes t ϱ + x ϱu ϱ x u = 0 t ϱu + x ϱu 2 ϱu x u = 0 t ϱe + x ϱeu ϱe x u = 0 Eigenvalues are given by u Usual CFL conditions for time-explicit schemes write t h max( u ) 1 2 The idea is then to propose a standard time-explicit scheme to keep accuracy on the slow phenomenon ( t = O(h) in all regime) 19/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

20 Operator splitting strategy We will consider the following three-step numerical scheme : First step (t n t Lag ) : solve implicitly the acoustic system with the solution at time t n as initial solution Second step (t Lag t n+1 ) solve implicitly the source terms when present with the solution at time t Lag as initial solution Third step (t n+1 t n+1 ) solve explicitly the transport system with the solution at time t n+1 as initial solution Solving implicitly the source terms avoid the time-step restriction t = O(ɛ) when ɛ << 1 ( t = O(h) in all regime) 20/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

21 A few words about the relaxation approach The gas dynamics equations in Lagrangian coordinates : with p = p(τ, e) and t τ m u = 0 t u + m p = 0 t E + m pu = 0 e = E 1 2 u2 Due to the nonlinearities of p, the Riemann problem is difficult to solve. The relaxation strategy : Idea : to deal with a larger but simpler system Design principle : to understand p(τ, e) as a new unknown that we denote Π 21/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

22 A few words about the relaxation approach The proposed relaxation system is At least formally, observe that t τ m u = 0 t u + m Π = 0 t E + m Πu = 0 t Π + a 2 m u = λ(p Π) lim Π = p (if a > ρc(τ, e)) λ + (see e.g. Chalons-Coulombel for a rigorous proof) Why is it interesting? The characteristic fields are linearly degenerate whatever the pressure law is! 22/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

23 A few words about the relaxation approach The time-explicit Godunov scheme applied to the relaxation system with initial data at equilibrium writes τ Lag u Lag Π Lag E Lag with Π n = p(τ n, en ) and = τ n + t m (u +1/2 u 1/2 ) = u n t m (p +1/2 p 1/2 ) = Π n a 2 t m (u +1/2 u 1/2 ) = E n t m (p +1/2 u +1/2 p 1/2 u 1/2 ) u +1/2 = 1 2 (un + u n +1) 1 2a (Πn +1 Π n ) p +1/2 = 1 2 (Πn + Π n +1) a 2 (un +1 u n ) 23/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

24 A few words about the relaxation approach The time-implicit Godunov scheme applied to the relaxation system with initial data at equilibrium writes τ Lag u Lag Π Lag E Lag with Π n = p(τ n, en ) and = τ n + t m (u +1/2 u 1/2 ) = u n t m (p +1/2 p 1/2 ) = Π n a 2 t m (u +1/2 u 1/2 ) = E n u +1/2 = 1 2 (ulag p +1/2 = 1 2 (ΠLag t m (p +1/2 u +1/2 p 1/2 u 1/2 ) + u Lag +1 ) 1 2a (ΠLag +1 ΠLag ) + Π Lag +1 ) a 2 (ulag +1 ulag ) 24/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

25 A few words about the relaxation approach The time-implicit scheme deals with (possibly strongly nonlinear) pressure laws is free of CFL restriction! is not expensive in the sense that only a linear problem w.r.t. u and Π has to be solved. Thanks to the relaxation strategy! 25/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

26 Formulation on unstructured meshes On unstructured meshes, the time-explicit ( = n) and time-implicit ( = Lag) schemes write u Lag = u n τ n Γ k t Ω Π k n k Π Lag τ Lag E Lag = Π n τ n t = τ n = E n + τ n t τ n t u k = 1 2 nt k (u +u k ) 1 2a k (Π k Π ), k N() k N() k N() k N() Γ k Ω (a k) 2 uk Γ k Ω u k Γ k Ω p k u k p k = 1 2 (Π +Π k ) a k 2 nt k (u k u ) 26/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

27 Source terms Introduction The time-implicit point-wise scheme for the gravity terms and external forces writes τ n+1 u n+1 E n+1 = τ Lag = u Lag = E Lag + t(g αu n+1 + t u n+1 ).(g αu n+1 ) It is free of CFL restriction 27/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

28 Transport step Introduction In order to approximate the solutions of the transport step t ρ + (u )ρ = 0 t (ρu) + (u )ρu = 0 t (ρe) + (u )ρe = 0 t ρ + (ρu) ρ u = 0 t (ρu) + (ρu u) ρu u = 0 t ρe + (ρeu) ρe u = 0 we simply use the time-explicit upwind finite-volume scheme ϕ n+1 = ϕ n+1 Γ k t k N() Ω u k ϕn+1 k where ϕ = ρ, ρu, ρe and ϕ n+1 k = ϕ n+1 + tϕ n+1 k N() if u k > 0 ϕ n+1 k if u k 0 Γ k Ω u k This scheme is stable under a material CFL condition ( t = O(h)) 28/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

29 Obectives Introduction Our obective was to propose a numerical scheme that is all-regime : uniform stability and uniform consistency w.r.t. ɛ able to deal with any equation of state multi-dimensional on (possibly) unstructured meshes What about the uniform consistency? 29/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

30 Uniform consistency in the low Mach regime Let us first recall that the Lagrangian-step in 1D writes with τ n+1 u n+1 E n+1 = τ n + t m (u +1/2 u 1/2 ) = u n t m (p +1/2 p 1/2 ) = E n t m ((pu) +1/2 (pu) 1/2 ) u +1/2 = 1 2 (u + u +1 ) 1 2a (p +1 p ) p +1/2 = 1 2 (p + p +1 ) a 2 (u +1 u ) 30/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

31 Uniform consistency in the low Mach regime In dimensionless form we get τ n+1 u n+1 E n+1 = τ n + t m (u +1/2 u 1/2 ) = u n t m (p +1/2 p 1/2 ) = E n t m ((pu) +1/2 (pu) 1/2 ) with, since p +1 p = O( mm 2 ), u +1/2 = u + u +1 2 p +1/2 = p + p +1 2M 2 a M (p +1 p ) 2a M 2 = u + u +1 +O(M m) 2 2M (u +1 u ) = p + p +1 2M 2 +O( m M ) 31/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

32 Uniform consistency in the low Mach regime The problem comes from the numerical diffusion in p +1/2 To obtain the uniform consistency w.r.t. M we introduce the parameter θ +1/2 and simply consider the new definition of p +1/2 p +1/2 = 1 2 (pn + p n +1) θ +1/2 a 2 (un +1 u n ) Then we get p+1/2 = p + p +1 2M 2 +O( θ +1/2 m M ) Which gives the uniform consistency provided that θ +1/2 = O(M) The modification is extremely simple and applies directly in multi-d 32/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

33 Uniform consistency in the large friction regime Let us first recall that the first two steps write τ n+1 u n+1 E n+1 with = τ n + t m (u +1/2 u 1/2 ) = u n t m (p +1/2 p 1/2 ) + t(g α ɛ un+1 = E n t m ((pu) +1/2 (pu) 1/2 ) u +1/2 = 1 2 (u + u +1 ) 1 2a (p +1 p ) ) + t un+1.(g α ɛ un+1 ) p +1/2 = 1 2 (p + p +1 ) a 2 (u +1 u ) 33/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

34 Uniform consistency in the large friction regime τ n+1 u n+1 = τ n + t m (u +1/2 u 1/2 ) = u n t m (p +1/2 p 1/2 ) + t(g α ɛ un+1 ) u +1/2 = 1 2 (un + u n +1) 1 2a (pn +1 p n ) p +1/2 = 1 2 (pn + p n +1) a 2 (un +1 u n ) Numerical asymptotic analysis. u = u (0) + ɛu (1) + O(ɛ 2 ) Multiply the second equation by ɛ and let ɛ 0 : u (0) = 0 Let ɛ 0 in the second equation : p +1 p 1 2 m It remains to insert u = 0 + ɛu (1) = (g αu(1) ) + O(ɛ 2 ) in the first equation 34/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

35 Uniform consistency in the large friction regime Let us insert u = 0 + ɛu (1) + O(ɛ 2 ) in the first equation, we immediately get with τ n+1 = τ n u (1) +1/2 = u(1) + u (1) +1 1 p +1 p 2 ɛ 2a + t m ɛ(u(1) +1/2 u(1) 1/2 ) + O(ɛ2 ) = u(1) + u (1) +1 + O( m 2 ɛ ) which is clearly not consistent with t τ ɛ m u 1 = O(ɛ 2 ) 35/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

36 Uniform consistency in the large friction regime The problem comes from the numerical diffusion in u +1/2 To obtain an uniform consistency w.r.t. ɛ we introduce the parameter θ +1/2 and simply consider the following definition of u +1/2 u +1/2 = 1 2 (un + u n +1) θ +1/2 2a (pn +1 p n ) Then we get u (1) +1/2 = u(1) + u (1) O( θ +1/2 m ) ɛ Which gives the uniform consistency provided that θ +1/2 = O(ɛ) The modification is extremely simple and applies directly in multi-d 36/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

37 Remarks Introduction All the obectives are reached! How does the modifications affect the stability properties? - conservative (with no source terms and external forces) - positive - unconditionally entropy satisfying for all θ 0 in the linear case - conditionally entropy satisfying in the non linear case. θ = 0 is also possible in practice! (numerical diffusion in the transport step) Interestingly, operator-splitting techniques are compatible with the all-regime property. USI approach not mandatory High-order extension under progress using DG methods, as well as shallow-water equations and diffusion terms 37/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

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58 Publications Introduction C. Chalons, M. Girardin and S. Kokh, Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms, SIAM J. Sci. Comput., 35(6) (2013) C. Chalons, M. Girardin and S. Kokh, Operator-splitting-based asymptotic preserving scheme for the gas dynamics equations with stiff source terms, AIMS on Applied Mathematics, Proceedings of the 2012 International Conference on Hyperbolic Problems : Theory, Numerics, Applications, 8 (2014) C. Chalons, M. Girardin and S. Kokh, An all-regime Lagrange-Proection like scheme for the gas dynamics equations on unstructured meshes, to appear in CICP (2016) C. Chalons, M. Girardin and S. Kokh, An all-regime Lagrange-Proection like scheme for 2D homogeneous models for two-phase flows on unstructured meshes, submitted to JCP M. Girardin, Méthodes numériques tout-régime et préservant l asymptotique de type Lagrange-Proection. Application aux écoulements diphasiques en régime bas Mach, Thèse de l Université Paris 6 (2014) 58/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

59 Numerical strategies Several approaches can be envisaged to compute accurate solutions when ɛ << 1 Use and discretize the limit model (the nature of which changes) Couple the original and limit models at moving interfaces Design Asymptotic-Preserving schemes (consistency with the limit model when ɛ 0 and with the original model when ɛ 0, no coupling) Consider all-regime stability and consistency properties (ɛ is kept constant in order to compute accurate solutions also in intermediate regimes) 59/59 Christophe Chalons All-regime Lagrangian-Remap numerical schemes