Finite difference MAC scheme for compressible Navier-Stokes equations

Transcription

1 Finite difference MAC scheme for compressible Navier-Stokes equations Bangwei She with R. Hošek, H. Mizerova Workshop on Mathematical Fluid Dynamics, Bad Boll May. 07 May. 11, 2018

2 Compressible barotropic Navier-Stokes ρ : density u : velocity p : pressure, p = aρ γ S : viscous stress, S = µ u, µ > 0 ρ t + div(ρu) = 0. (ρu) t + div(ρu u) = p + S (1a) (1b) Boundary condition for u u Ω = 0 or periodic (1c) Initial values ρ(x, 0) = ρ 0 > 0 (1d) Motivation: stability and convergence 2 / 22 B. She FD-MAC scheme for compressible NS

3 Literature Finite Volume-Finite Element by T. Karper, 2013, γ > 3 E. Feireisl, R. Hošek, D. Maltese, A. Novotný, 2017 bounded numerical solution E. Feireisl, M. Lukáčová-Medvid ová, 2017 dissipative measure-valued solution, γ (1, 2) Our interests: Finite Difference 3 / 22 B. She FD-MAC scheme for compressible NS

4 Notations I Elements: Ω h = K Faces: E Exterior faces: E ext = Ω E. Interior faces:e int = E \ E ext Faces of element K: E(K) Primary grid o : density, pressure Dual grid : velocity E(K) := { σ = K ± 1 } 2 e s, K Ω h, s = 1,, d, σ,s± = K ± 1 2 e s. 4 / 22 B. She FD-MAC scheme for compressible NS

5 Notations II Between grids {f } σ = 1 2 (f K + f L ), σ = K L E int. ḡ K = 1 gσ, g 1 σ,1 g σ, g σ,2 2. gσ, g σ,3 3 The s-th component of vector g is defined on the face σ E. Functional spaces We denote the space for P0 functions with respect to the grid Ω h by X (Ω h ) = {f L (Ω); f K f K R}. The s-th component of g is constant in the neighbourhood of the edge, not in the element K. X (E int ) d = { g X (E) d ; g Eext = 0 } 5 / 22 B. She FD-MAC scheme for compressible NS

6 Discrete differential operators I Time t h φn = φn φ n 1 t ( h f ) K = 1 h 2 Space Let σ = K L, L = K + e s, s = 1,, d. ( s hf ) σ = f L f K h L N (K) (f L f K ), ( h g) σ = 1 d h 2 (g σ es 2g σ + g σ+es ). s=1, f X (Ω h ), 6 / 22 B. She FD-MAC scheme for compressible NS

7 Discrete differential operators II Upwind flux Let f + = max{0, f }, f = min{0, f }. The upwind flux is given by Up[f, u] σ = f K (u s σ) + + f L (u s σ), Upwind discrete derivative and upwind divergence Up s [f, u] K = Up[f, u] σ,s+ Up[f, u] σ,s, h d div Up [g, u] K = s Up [f, u] K. s=1 Let f X (Ω h ), v = [v 1,, v d ] X (E int ) d, then div Up [f, v] K = 0. K Ω K h 7 / 22 B. She FD-MAC scheme for compressible NS

8 Numerical Scheme t hρ n K + div Up [ρ n, u n ] K h α ( h ρ n ) K = 0, (2a) { h(ρū) t n } σ + {div Up [ρ n ū n, u n ]} σ + ( hp(ρ s n ) ) e σ s µ ( h u n) d σ { ( hα r h {û n } hρ r n)} = 0, (2b) σ for all K Ω h, σ E int and n = {1,..., N}, with boundary conditions. r=1 8 / 22 B. She FD-MAC scheme for compressible NS

9 Renormalized continuity equation Lemma 1 Let (ρ h, u h ) satisfy the discrete density equation (2a). Then for any B C 2 (R) it holds ( ( ) hb(ρ t n K ) + B (ρ n K )ρ n K B(ρ n K ) )(div h u n ) K + P K = 0, (3) where K Ω h P K = t 2 B (ρn 1,n K ) t hρ n K σ E(K) ( h u σ + h α) B (ρ σ ) ( h ρ) σ 2 Note that P K 0 provided B is convex. 9 / 22 B. She FD-MAC scheme for compressible NS

10 Lemma 2 (Existence of num sol) Let ρ n 1 h X (Ω h ), u n 1 h X (E int ) d be given; ρ n 1 K > 0 for all K Ω h. Then the numerical scheme (2a-2b) admits a solution ρ n h X (Ω h ), ρ n K > 0 for all K Ω h, u n h X (E int ) d. Moreover, is satisfies the discrete conservation of mass ρ n K = ρ n 1 K. K Ω h K K Ω h K Mass conservation Positivity : non-negativity + strictly positivity 10 / 22 B. She FD-MAC scheme for compressible NS

11 Positivity Recall the renormalized continuity equation (3) with test function { ( z) η for z < 0, B(z) = η for z 0. B (z)z B(z) = 0. B(ρ n K ) = ( B(ρ n 1 K ) P ) K 0. K Ω h K Ω h K K K Ω h max{ ρ n K, 0} 0. ρ n K 0. Choose K Ω h such that ρ n K ρn L for all L Ω h. Then we have ρ n K ρ n 1 K = tdiv Up [ρ n, u n ] K + th α ( h ρ n ) t d ( ) ρ n K uσ s h s,+ ρ n K uσ s s, + (ρ n K+e s ρ n K )uσ s s,+ + (ρ n K ρ n K e s )uσ s + s, s=1 tρ n K (div h u n ) K tρ n K (div h u n ) K. ρ n L ρ n K t (div h u n ) K ρn 1 K > 0, for any L Ω h, K 11 / 22 B. She FD-MAC scheme for compressible NS

12 Lemma 3 (Energy stability) Let (ρ h, u h ) be the numerical solution obtained through the scheme (2a 2b). For any time step m = 1,, N the following estimate holds, K Ω h K ( ρ m ū m K 2 K ) γ 1 p(ρm K ) + tµ + 4 N j j=1 K Ω h m n=1 K Ω h K 3 K r=1 s=1 m d 1 N 1 = t n=1 K Ω K 2 h s=1 m N 2 = t 2 p (ρ η K ) m h t n=1 K Ω K 2 ρn K 2, N 3 = t 2 h n=1 K Ω K h N 4 = t h m Up[ρ n, u n ] σ ( s 4 n=1 Γ E Γ hūn ) σ 2. int 3 ( h(u r s ) n ) K 2 ( ρ 0 ū 0 K 2 K + 1 ) 2 γ 1 p(ρ0 K ), ( (h α + h 2 (u s,n σ,s )± )p (ρ n, σ,s ) ( s h ρn ) σ,s 2 ), ρ n 1 K 2 h t ūn K 2, 12 / 22 B. She FD-MAC scheme for compressible NS

13 Lemma 4 (Uniform bounds) Let (ρ h, u h ) be a numerical solution obtained through the scheme (2a) (2b) and let the total initial energy E 0 be defined by 1 E 0 = 2 ρ 0u γ 1 p(ρ 0)dx. Ω Then there exists a constant c only depend on E 0 (independent of h, t), such that ρ h L (L γ (Ω)) c. p(ρ h ) L (L 1 (Ω)) c. h u h L 2 (L 2 (Ω)) c. u h L 2 (L 6 (Ω)) c. ρ h ū h L (L 2 (Ω)) c. ρ h L 2 (L (Ω)) ch θ 1, θ = θ(α, γ) > / 22 B. She FD-MAC scheme for compressible NS

14 Lemma 5 (Consistency for continuity ) Let ρ n h, ûn h be piecewise constant and piecewise affine representations, respectively, of the solution to the numerical scheme (2a 2b). Then for any φ C 1 (Ω) it holds that t h ρ n hφdx ρ n hûn h x φdx = h β R h x φdx, (4) Ω Ω Ω where β > 0 and R h L 1 (0,T ;L 3/2 (Ω)) 1. Multiply (2a) with (Π P φ) K. 14 / 22 B. She FD-MAC scheme for compressible NS

15 Lemma 6 (Consistency for momentum) Let (ρ n h, un h ) be piecewise constant representations of the solution to numerical scheme (2a 2b). Then for any v C0 1(Ω) W 2,q (Ω), q > 1 it holds that h(ρ t h ū h ) n vdx Ω + µ Ω Ω ρ n hūn h ū n h : x vdx p(ρ n h)div x vdx Ω ( h u n h) : x vdx = h θ1 r 1 h, x v + h θ2 r 2 h, 2 xv, (5) where r 1 h L1 (0, T ; L 6γ 5γ+6 (Ω)), r 2 h L 1 (0, T ; L q (Ω)), θ 1, θ 2 > 0. Multiply momentum scheme (2b) with Π D v. 15 / 22 B. She FD-MAC scheme for compressible NS

16 Definition 7 (Dissipative measure-valued solution) We say that a parameterized measure {ν t,x } (t,x) (0,T ) Ω, ( ( ) ) ν L weak (0, T ) Ω; P [0, ) R N ; ν t,x ; s ρ, ν t,x ; v u, is a dissipative measure-valued solution of the Navier-Stokes system in (0, T ) Ω, if the following holds for a.a. τ (0, T ), for any ψ C 1( (0, T ) Ω; R d ) [ ] t=τ τ τ ν τ,x ; s ψ(τ, )dx = [ ν t,x ; s t h ψ + νt,x ; sv x ψ]dxdt + r C ; x ψ dt, Ω t=0 0 Ω 0 [ ] t=τ τ ν τ,x ; sv ψ(τ, )dx = [ ν t,x ; sv t hψ + νt,x ; sv v : x ψ + νt,x ; p(s) ]dxdt Ω t=0 0 Ω τ τ S( u) : x ψdxdt, + r M ; x ψ dt 0 Ω 0 [ ν τ,x ; 1 Ω 2 s v 2 + p(s) ] t=τ τ ψ(τ, ) x + S( u) : x ψdxdt + D(τ) 0, γ 1 t=0 0 Ω where r C L 1( [0, T ]; M( Ω) ), r M L 1( [0, T ]; M( Ω) ) satisfying r C (τ); ψ χ(τ)d(τ) ψ C( Ω), r M (τ); ψ ξ(τ)d(τ) ψ C( Ω), for χ, ξ L1 (0, T ). In addition, for a.a. τ (0, T ) it holds τ ν τ,x ; v u 2 dxdt c P D(τ) / 22 B. She FD-MAC scheme for compressible NS

17 Theorem 8 Let [ρ k h, uk h ]N T k=1 be a family of numerical solutions obtained by scheme (2) with 1 < γ < 2, δt h, 1 < α < 2γ 1 and suppose that the initial data satisfy ρ 0 L (R d ), ρ 0 ρ > 0 a.a. in R d, u 0 L 2 (R d ). Then any Young measure {ν t,x t,x (0,T ) Ω } generated by ρ k h, uk h represents a dissipative measure-valued solution of the Navier-Stokes system (1) in the sense of Definition 7. Applying the weak-strong uniqueness 1 we conclude Theorem 9 (Convergence) In addition the hypotheses of Theorem 8, suppose that the Navier-Stokes system (2) endowed with the initial data [ρ 0, u 0 ] and periodic boundary condition admits a regular solution [ρ, u] belonging to the class ρ, x ρ, u, x u C([0, T ] Ω, t h u L2 ( 0, T ; C( Ω); R d ) ), ρ > 0, u Ω = 0. Then for any compact K Ω. ρ h ρ (strongly) in L γ ((0, T ) K), u h u (strongly) in L 2 ( (0, T ) K; R d ) 1 Feireisl et.al. Dissipative measure-valued solutions to the compressible Navier Stokes system. Calc. Vari. Partial Differ. Equ / 22 B. She FD-MAC scheme for compressible NS

18 Numerical test Ω = [0, 1] 2, µ = 0.01, a = 1.0, γ = 1.4, α = t = CFL h u max Cavity flow, upper boundary u = (16x 2 (1 x) 2, 0) T. Table : Convergence results of cavity flow h u L 2 (L 2 ) EOC u L 2 (L 2 ) EOC ρ L 1 (L 1 ) EOC ρ L (L γ ) EOC 1/ e e e e-02 1/ e e e e / e e e e / e e e e / 22 B. She FD-MAC scheme for compressible NS

19 (a) density ρ (b) velocity U (c) velocity V 19 / 22 B. She FD-MAC scheme for compressible NS

20 (a) density ρ (b) velocity U (c) velocity V 20 / 22 B. She FD-MAC scheme for compressible NS

21 U(0, x, y) = u r (r) (y 0.5)/r, V (0, x, y) = u r (r) (0.5 x)/r. where r = (x 0.5) 2 + (y 0.5) 2 and u r (r) = 2r/R if 0 r < R/2, γ 2(1 r/r) if R/2 r < R, 0 if r R, Table : Convergence results of Gresho vortex test h u L 2 (L 2 ) EOC u L 2 (L 2 ) EOC ρ L 1 (L 1 ) EOC ρ L (L γ ) EOC 1/ e e e e-03 1/ e e e e / e e e e / e e e e / 22 B. She FD-MAC scheme for compressible NS

22 Future work A penalty method (ρu) t + div(ρu u) + 1 ε 1 Ω s u = p + S { 1 x Ωs 1 Ωs = 0 x Ω f Thank you for your attention! 22 / 22 B. She FD-MAC scheme for compressible NS