Convergence of the MAC scheme for incompressible flows

Transcription

1 Convergence of the MAC scheme for incompressible flows T. Galloue t?, R. Herbin?, J.-C. Latche??, K. Mallem????????? Aix-Marseille Universite I.R.S.N. Cadarache Universite d Annaba

2 Calanque de Cortiou Marseille main sewer

3 Incompressible Variable density flows Bounded domain (0, T ), R d, T > 0. Non homogeneous fluid : density ρ is not constant Incompressible fluid : div ū = 0, Incompressible variable density Navier-Stokes equations: t ρ + div( ρū) = 0, t ρ ū + div( ρ ū ū) ū + p = 0, div ū = 0, u = 0, u t=0 = u 0, ρ t=0 = ρ 0 ρ min > 0, (VDNS)

4 Variable density NS equations: a priori estimates maximum principle for ρ divū = 0 and t ρ + div( ρū) = 0 t ρ + ū ρ = 0 (transport equation) 0 < min ρ 0 ρ max ρ 0. kinetic energy balance: Multiply momentum balance eq by ū and use mass balance (twice) t ( 1 2 ρ ū 2 ) + div( 1 2 ρ ū 2 u) ū ū + p ū = 0. L ((0, T ); L 2 () d ) and L 2 ((0, T ); H 1 0 ()d ) bound: 1 t ρ(., t) ū(., t) 2 dx ū 2 dx dt = 1 ρ 0 ū 0 2 dx, t (0, T ). 2

5 Variable density NS equations: weak solutions Weak Solutions: ρ 0 L () such that ρ 0 > 0 and u 0 L 2 () d. A weak solution is a pair (ρ, u) such that: ρ L ((0, T ) ) and ρ > 0 a.e. in (0, T ). ū L ((0, T ); L 2 () d ) L 2 ((0, T ); H 1 0 ()d ) et divū = 0 a.e. in (0, T ). For any φ in C c ( [0, T )), T ( ρ t φ + ρū φ) dx dt = ρ 0 φ(., 0) dx. 0 for any v in C c ( [0, T ))d such that divv = 0, T ( ) ρū t v ( ρū ū) : v + ū : v dx dt = ρ 0 ū 0 v(., 0) dx. 0 Existence of a weak solution Simon 90. Discretization Liu Walkington Discontinuous Galerkin 2007 Latché Saleh Rannacher Turek 2016

6 The standard Marker-And-Cell scheme (Harlow Welsh 65) structured grids, pressure at the cell-center and normal velocity components at mid-edges, staggered meshes. Advantages and drawback + minimal number of unknowns, + no pressure stabilization needed, + simplicity and robustness of the scheme, needs local refinement for complex geometries. Convergence analysis: Shin Strickwerda 96, 97, stability, inf-sup condidion Nicolaides Wu 96, (ω, ψ) formulation Blanc 05 irregular rectangles, finite volume approach

7 The MAC grid (or Arakawa C) Primal mesh (pressure) T : K : Primal cell. E set of faces of T. E (i) ; i = 1,..., d : set of faces orthogonal to the i-th component of e i. Dual mesh (velocity) : Dσ dual cell associate to the face σ E (i). D σ K D σ D σ ρ = K T ρ K χ K, u (i) = σ E (i) u σχ Dσ

8 From the continuous to the discrete equations: time-implicit discretization Continuous equations ρ L ((0, T ) ), ū L ((0, T ); L 2 () d ) L 2 ((0, T ); H 1 0 ()d ), div ū = 0, t ρ + div( ρū) = 0, t ( ρ) ū + div( ρ ū ū) ū + p = 0, Discrete equations ρ (n+1), p (n+1) L T, u (n+1) H E, div T u (n+1) = 0, ð t ρ (n+1) T + div T ((ρu) (n+1) ) = 0 ð t (ρu) (n+1) + c E ((ρu) (n+1) )u (n+1) E u (n+1) + E p (n+1) = 0, Discrete time derivative: ð t (φ) (n+1) = 1 δt (φ(, t n+1) φ(, t n)), φ H T or φ H E. Discrete operators: div T u, div T ρu, c E ((ρu))u, E u, E p...

9 The time-implicit MAC scheme Discrete equations ρ, p piecewise constant on the primal mesh, u piecewise constant on the dual mesh ρ (n+1), p (n+1) L T, u (n+1) H E, div T u (n+1) = 0, ð t ρ (n+1) T + div T ((ρu) (n+1) ) = 0 ð t (ρu) (n+1) + c E ((ρu) (n+1) )u (n+1) E u (n+1) + E p (n+1) = 0, Discrete operators div T u, div T (ρu) : For K T : div K u (n+1) = 1 K 1 δt (ρ(n+1) K σ E(K ) ρ (n) K ) + 1 K σ u K,σ = 0, with u K,σ = ±u σ, σ E(K ) with F K,σ = σ ρ up σ u K,σ, F (n+1) K,σ = 0, ρ up σ : upwind approximation of ρ on σ = ρ(n+1) > 0. K

10 The time-implicit MAC scheme Discrete equations, ρ, p piecewise constant on the primal mesh, and u piecewise constant on the dual mesh ρ (n+1), p (n+1) L T, u (n+1) H E, div T u (n+1) = 0, ð t ρ (n+1) T + div T ((ρu) (n+1) ) = 0 ð t (ρu) (n+1) + c E ((ρu) (n+1) )u (n+1) E u (n+1) + E p (n+1) = 0, For σ E : 1 δt (ρ(n+1) σ u σ ρ (n) σ u(n) σ ) + 1 F (n+1) σ,ɛ D σ u(n+1) ɛ ( u) (n+1) σ ɛ Ē(Dσ ) ρ σ and F σ,ɛ chosen later. centered approximation for u ɛ, with: ( u) σ = two point flux FV approximation on the velocity meshes, E p (n+1) = ( p) σ, ( p) σ = σ D (p L p K )n K,σ, σ E σ int K D σ ɛ = σ σ + ( p) (n+1) σ σ σ = K L ɛ == σ σ σ = 0, σ E int. L

11 Discrete duality, coercivity, inf-sup property Discrete divergence-gradient duality q div T v + E q v = 0. Coercivity d u 1,E = D σ E (i) u (i) u σ = E u L 2 () d C u L 2 () d i=1 σ E (i) Inf-sup property q L T, v H E ; q = div T v and v E C p L 2 () d (discrete version of Necas Lemma, Strickwerda 90, Blanc 05)

12 Discrete kinetic energy inequality, choice of ρ σ and Discrete equations If a mass balance holds on the dual cells: D σ 2δt (ρ σ ρ σ ) + σ E F σ,ɛ = 0, and ρ σ > 0, then 1 ( ρ σuσ 2 ρ 2δt σ(uσ) 2) D σ Proof Choice of ρ σ and F σ,ɛ ɛ Ẽ(D σ) ɛ=σ σ F σ,ɛu σu σ ( u) σu σ+(ðp) σu σ 0 Herbin, Kheriji, Latché 2013 τ ɛ τ F σ,ɛ = D σ ρ σ = D K,σ ρ K + D L,σ ρ L, 1 2 (F K,τ + F K,τ ) if ɛ σ, ɛ τ τ 1 2 ( F K,σ + F K,σ ) if ɛ K = [σ, σ ] K ɛ K σ = K L D σ L

13 Convergence analysis Theorem : (T m, E m), δt m sequence of meshes and time steps h m 0, δt m 0 + suitable assumptions on the mesh, There exists ρ m, u m solution to the scheme, and, up to a subsequence ρ m ρ in L p ( (0, T )), p [1, + [, u m ū in L 2 ( (0, T )) ( ρ, ū) is a weak solution to (VDNS) Sketch of proof A priori estimates on the approximate solutions ρ m, u m existence of a solution to the scheme. Compactness thanks to discrete functional analysis tools: Up to a subsequence, ρ m ρ, u m ū for some norms. Passage to the limit in the weak form of the scheme, ( ρ, ū) is a weak solution to (VDNS).

14 Convergence analysis: estimates and existence 1 - A priori estimates. Maximum principle (upwind discretization of the mass balance): min ρ 0(x) ρ max ρ 0(x). x x Kinetic energy identity + grad-div duality: u L ((0,T );L 2 () d ) + u L 2 ((0,T );H 1 E,0 ) C Estimate of the momentum balance convection form: C E (ρu)v w C ηt ρ L () u L 4 () d v L 4 () d w 1,E,0 C ηt ρ L () u 1,E,0 v 1,E,0 w 1,E, Existence of a solution thanks to a topological degree argument.

15 Easy case : homogeneous fluid, constant density Estimate on u m in L 2 (0, T ; H 1 E,0 ) Estimate on ð t u m N 1 ð t u m L 1 (0,T ;E ) = δt ð t u E C where v E = max E E E ϕ E v ϕ dx ; n=0 E ϕ 1,E,0 1 Discrete Aubin-Simon theorem = compactness on u m in L 2 ((0, T ), L 2 ()) Discrete Aubin-Simon theorem 1 p < +, B Banach space, B m B, dim B m < +, Xm and Ym norms on B m s.t. : if w m Xm is bounded then: (i) w B; w m w in B up to a subsequence, (ii) If w m w in B, and w m Ym 0 then w = 0. Let T > 0 and (u m) m N be a sequence of L p ((0, T ), B), piecewise constant on the time intervals, such that 1. ( T 0 um p Xm dt) m N is bounded, 2. ( T 0 ðum ym dt) m N is bounded. Then there exists u L p ((0, T ), B) such that, up to a subsequence, u m u in L p ((0, T ), B). u m ū L 2 ((0, T ), L 2 ()) + (u m) m N bounded in L 2 ((0, T ), H 1 E,0 ) ū L 2 ((0, T ), H 1 0 ()).

16 Variable density, estimate on the translates No estimate on ð t u m Estimate on the time translates (continuous case: Boyer Fabrie 13 chapter 2); T τ u m(x, t + τ) u m(x, t) 2 ρ max dx dt C ηt,t ( u m 3 0 ρ L 2 (H min E,0 ) + 1) τ + δt Kolmogorov compactness theorem compactness of u m in L 2 ((0, T ), L 2 ()). τ + δt : No estimate on time derivative. Will not generalize to the compressible case because of ρ min

17 Variable density, convergence Convergence proof, sketch : u m ū in L 2 (0, T ; L 2 () d ), ρ m ρ weakly in L 2 (0, T ; L 2 ()) T T ρ m u m ψ ρ ū ψ for any ψ L ( (0, T )) d ( ) 0 0 In particular for ψ = mϕ, T T div m(ρ m u m) ϕ dxdt = ρ m u m mϕ dxdt ( ρ ū) ϕ dxdt 0 0 (thanks to weak BV estimate) ð t ρ m t ρ, ð t (ρ m u m) t (ρu), divu m divū div mū = 0 (linear operators) ρ mu m bounded in L 2 (0, T ; L 2 () d ) = ρ mu m converges weakly in L 2 (0, T ; L 2 () d ). Hence by ( ), ρ mu m ρū weakly in L 2 (0, T ; L 2 () d ) and since u m u in L 2 (0, T ; L 2 () d ) ρ mu m u m ϕ ρū u ϕ

18 Variable density case, convergence, alternate proof Compactness of ρ m in L 2 (H 1 ) thanks to Aubin-Simon with B = H 1 () X = L 2 () Y = W 1,1 () X = L 2 () compactly embedded in B = H 1 () t ρ m bounded in L 1 (0, T, W 1,1 ()) ρ m bounded in L 2 (0, T, L 2 ()) ρ m compact in L 2 (0, T, H 1 ()) u m bounded in L 2 (0, T, H 1 0 ()) hence (weak strong product) ρ mu m ρū weakly in L 1. Same ideas for the convergence of ρ mu m u m... Thierry Gallouët, 2017

19 Recent and on going work Locally refined grids : with Chénier, Eymard, Gallouët for modified MAC, Latché, Piar, Saleh for RT grids. Stability and weak consistency for the full NS equations (with D. Grapsas, W. Kheriji, J.-C. Latché) Convergence for steady-state barotropic NS (γ 3) (with J.-C. Latché, T. Gallouët and D. Maltese) Error estimates for barotropic NS, adaptation of strong weak uniqueness (with D. Maltese, and A. Novotny). Stability and weak consistency for the Euler equations with higher order schemes and entropy consistency (with T. Gallouët, J.-C. Latché, N. Therme) Low Mach limit for barotropic NS (with J.-C. Latché, K. Saleh). Convergence of the fully discrete pressure correction scheme. TODO Convergence for the time-dependent barotropic compressible Navier-Stokes. TODO