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

Transcription

1 All-regime Lagrangian-Remap numerical schemes for the gas dynamics equations. Applications to the large friction and low Mach regimes Christophe Chalons LMV, Université de Versailles Saint-Quentin-en-Yvelines Joint works with M. Girardin and S. Kokh 1/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

2 Outline Introduction 1 Introduction /63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

3 Outline Introduction 1 Introduction /63 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 4/63 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 in the following three flow regimes Classical regime : ɛ = O(1) Low ɛ regime : ɛ << 1 Limit regime : ɛ 0 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 These requirements will be specified later on... 5/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

6 Outline Introduction 1 Introduction /63 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 7/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

8 Large friction regime Setting in t ρ + (ρu) = 0 u = u 0 + ɛu 1 + O(ɛ 2 ) t (ρu) + (ρu u) + p = ρ(g α ɛ u) t (ρe) + [(ρe + p)u] = ρu.(g α ɛ u) the behaviour of the solutions is given by u 0 = 0 t ρ + ɛ (ρu 1 ) = O(ɛ 2 ) p = ρ(g αu 1 ) t (ρe) + ɛ [(ρe + p)u 1 ] = ɛρu 1.(g αu 1 ) + O(ɛ 2 ) 8/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

9 Large friction regime Note that replacing t with t ɛ in u 0 = 0 t ρ + ɛ (ρu 1 ) = O(ɛ 2 ) p = ρ(g αu 1 ) t (ρe) + ɛ [(ρe + p)u 1 ] = ɛρu 1.(g αu 1 ) + O(ɛ 2 ) the long time behaviour is given by u 0 = 0 t ρ + (ρu 1 ) = O(ɛ) p = ρ(g αu 1 ) t (ρe) + [(ρe + p)u 1 ] = ρu 1.(g αu 1 ) + O(ɛ) see Hsiao-Liu, Nishihara, Junca-Rascle, Lin-Coulombel, Coulombel-Goudon, Marcati-Milani... for rigorous proofs 9/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

10 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 t ρ + (ρu) = 0 t (ρu) + (ρu u) + 1 t (ρe) + [(ρe + p)u] + M2 2 where M = u 0 c 0 M 2 p ( = 0 t (ρu.u) + (ρu.uu) ) = 0 denotes the Mach number and plays the role of ɛ 10/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

11 Low Mach regime Introduction t ρ + (ρu) = 0 t (ρu) + (ρu u) + 1 t (ρe) + [(ρe + p)u] + M2 2 M 2 p ( = 0 t (ρu.u) + (ρu.uu) ) = 0 Remark 1. The flow is said to be in the low Mach regime if M << 1 and p = O(M 2 ) Remark 2. Using asymptotic expansions of ρ, u, p, c in powers of M in the governing equations of ρ, u, p, together with boundary conditions on a given domain D (global argument), we get t ρ 0 + (ρ 0 u 0 ) = 0 t u 0 + (u 0 )u ρ 0 p 2 = 0 u 0 = 0 11/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

12 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 12/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

13 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) 13/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

14 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) 14/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

15 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 15/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

16 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... 16/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

17 Outline Introduction 1 Introduction /63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

18 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 18/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

19 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 9/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

20 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 also writes t τ m u = 0 t u + m p = 0 t E + m pu = 0 with τ = 1/ϱ and τ x = m. 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) 20/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

21 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) 21/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

22 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) 22/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

23 A few words about the relaxation approach The gas dynamics equations in Lagrangian coordinates : t τ m u = 0 t u + m p = 0 t E + m pu = 0 with p = p(τ, e) and 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 23/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

24 A few words about the relaxation approach The gas dynamics in Lagrangian coordinates t τ m u = 0 t u + m p = 0 t E + m pu = 0 The relaxation system t τ m u = 0 t u + m Π = 0 t E + m Πu = 0 t Π + a 2 m u = λ(p Π) At least formally, observe that lim Π = p (if a > ρc(τ, e)) λ + (see e.g. Chalons-Coulombel for a rigorous proof) 4/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

25 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 ) 25/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

26 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 ) 26/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

27 A few words about the relaxation approach The time-explicit scheme deals with (possibly strongly nonlinear) pressure laws is stable and satisfies a discrete entropy inequality provided that a is chosen sufficiently large and under a CFL restriction t m max(ρc) 1 2 In dimensionless form (low Mach regime), it writes t m max(ρ c M ) 1 2 that is to say t = O(hM) 27/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

28 A few words about the relaxation approach The time-implicit scheme deals with (possibly strongly nonlinear) pressure laws is free of CFL restriction! is cheap in the sense that only a linear problem w.r.t. u and Π has to be solved In 1D, the following two equations are decoupled { t (Π + au) + a x (Π + au) = 0 t (Π au) a x (Π au) = 0 28/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

29 Formulation on unstructured meshes On unstructured meshes, the time-explicit ( = n) and time-implicit ( = Lag) schemes write = u n τ n t u Lag 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 Π k n k Γ k Ω (a k) 2 u k 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 ) 29/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

30 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 0/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

31 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 t k N() Γ k Ω u k ϕn+1 k where ϕ = ρ, ρu, ρe and ϕ n+1 k = + tϕ n+1 k N() { ϕ n+1 if uk > 0 ϕ n+1 k if uk 0 Γ k Ω u k This scheme is stable under a material CFL condition ( t = O(h)) 31/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

32 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 What about the first obective? 32/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

33 Uniform consistency in the large friction regime Let us first focus on the first two steps of the time-explicit scheme (the transport step is not a problem) 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 u n+1 t m ( (pu) +1/2 (pu) 1/2 u +1/2 = 1 2 (u + u +1 ) 1 2a (p +1 p ) ).(g α ɛ un+1 ) p +1/2 = 1 2 (p + p +1 ) a 2 (u +1 u ) 33/63 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 = (g αu (1) ) 2 m Let then insert u = 0 + ɛu (1) + O(ɛ 2 ) in the first equation : 34/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

35 Uniform consistency in the large friction regime τ n+1 = τ n + t m ɛ(u(1) +1/2 u(1) 1/2 ) + O(ɛ2 ) u +1/2 = 1 2 (un + u n +1) 1 2a (pn +1 p 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 Let then ɛ 0 in the first equation : u (1) +1/2 = u(1) + u (1) +1 m p +1 p 2 ɛ 2a m = (g αu (1) ) = u (1) + u (1) +1 + O( m 2 ɛ ) which is clearly not consistent with t τ ɛ m u 1 = O(ɛ 2 ), 35/63 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 In order to obtain an uniform consistency with respect to 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) 36/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

37 Uniform consistency in the low Mach regime Let us focus on the first step of the time-explicit scheme (the transport step is not a problem) 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 ( (pu) m +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 ) 37/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

38 Uniform consistency in the low Mach regime In dimensionless form we get τ n+1 u n+1 E n+1 with, since p +1 p = O(M 2 ) u +1/2 = u + u +1 2 p +1/2 = p + p +1 2M 2 = τ n + t m (u +1/2 u 1/2 ) = u n t m (p +1/2 p 1/2 ) = E n t ( (pu) m +1/2 (pu) ) 1/2 M m (p +1 p ) 2aM 2 m a m 2M (u +1 u ) m = u + u +1 +O(M m) 2 = p + p +1 2M 2 +O( m M ) 38/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

39 Uniform consistency in the low Mach regime The problem comes from the numerical diffusion in p +1/2 In order to obtain an uniform consistency with respect to M we introduce the parameter θ +1/2 and simply consider the following 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) 39/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

40 Remarks Introduction The modifications give the uniform consistency and we recover the classical scheme provided that θ +1/2 = 1 The modifications apply directly on unstructured meshes Considering the time-implicit treatment of the Lagrangian step gives the uniform stability The relaxation approach allows to consider any given pressure law Recall that the unstructured mesh is fixed (not moving) All the obectives are reached 40/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

41 Remarks Introduction Interestingly, we proved that operator splitting strategies are compatible with asymptotic-preserving and all-regime properties! How does the modifications affect the stability properties? One are able to prove that the schemes are - conservative (with no source terms and external forces) - positive - uniformly stable and uniformly consistent w.r.t. ɛ - entropy satisfying under a suitable definition of θ θ = 0 is also possible! (numerical diffusion in the transport step) High-order extension under progress using DG methods 41/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

42 Outline Introduction 1 Introduction /63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

43 Implementation of the numerical scheme The numerical schemes have been implemented in YAFiVoC (Yet Another Finite Volume Code) A code that was developed by Mathieu Girardin and Samuel Kokh to implement finite volume methods on unstructured meshes Programming Language : C Compilation : CMake Linear problem solver : Petsc 43/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes

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63 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, submitted to CICP 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) 63/63 Christophe Chalons All-regime Lagrangian-Remap numerical schemes