On Godunov type schemes accurate at any Mach number ephane - PowerPoint PPT Presentation

CANUM 2011, Guidel, France . On Godunov type schemes accurate at any Mach number ephane Dellacherie 1 , 3 St´ In collaboration with P. Omnes 1 , 3 and P.A. Raviart 2 , 3 1 CEA-Saclay 2 Universit´ e Paris 6 3 LRC-Manon, LJLL, Paris 6 May 25 th , 2011 . On Godunov type schemes accurate at any Mach number 1 .

. Outline Introduction I - The low Mach number problem and the linear wave equation II - The linear case at any Mach number III - The non-linear case at any Mach number . On Godunov type schemes accurate at any Mach number 2 .

. Introduction I - The low Mach number problem and the linear wave equation II - The linear case at any Mach number III - The non-linear case at any Mach number . On Godunov type schemes accurate at any Mach number 3 .

Introduction When M := u ∗ a ∗ ≪ 1 and when the initial conditions are well-prepared in the sense .  ρ ( t = 0 , x ) = ρ ∗ ( x ) ,  p ( t = 0 , x ) = p ∗ + O ( M 2 ) ,  u ( t = 0 , x ) = � u ( x ) + O ( M ) with ∇ · � u ( x ) = 0 , the solution ( ρ, u , p ) of the (dimensionless) compressible Euler system  ∂ t ρ + ∇ · ( ρ u ) = 0 ,       ∂ t ( ρ u ) + ∇ · ( ρ u ⊗ u ) + ∇ p M 2 = 0 ,       ∂ t ( ρ E ) + ∇ · [( ρ E + p ) u ] = 0 is close to ( ρ, u , p ) which satisfies the incompressible Euler system  ∂ t ρ + u · ∇ ρ = 0 , ρ ( t = 0 , x ) = ρ ∗ ( x ) ,     ∇ · u = 0 and u ( t = 0 , x ) = � u ( x ) ,     ρ ( t , x )( ∂ t u + u · ∇ u ) = −∇ Π (with variable density when ρ ′ ∗ ( x ) � = 0) and p = p ∗ . Idem for the Navier-Stokes syst. when the thermal fluxes are not high. . On Godunov type schemes accurate at any Mach number 4 .

Introduction . Nevertheless, when we apply a (2D or 3D) Godunov type scheme on a mesh that is not triangular, the discrete compressible Euler solution: ◮ converges with high difficulties to an incompressible solution when ∆ x → 0 ( M ≪ 1 is given); ◮ does not converge to an incompressible solution when M → 0 (∆ x is given). . On Godunov type schemes accurate at any Mach number 5 .

Introduction For example, we find in [Guillard et al. , 1999] when the mesh is not triangular: . . On Godunov type schemes accurate at any Mach number 6 .

Introduction . Nevertheless, when the mesh is TRIANGULAR, the results seem to remain accurate: Iso-Mach, VFRoe scheme, M = 10 − 2 Iso-press., VFRoe scheme, M = 10 − 2 WHAT HAPPENS !?!? . On Godunov type schemes accurate at any Mach number 7 .

. Introduction I - The low Mach number problem and the linear wave equation II - The linear case at any Mach number III - The non-linear case at any Mach number . On Godunov type schemes accurate at any Mach number 8 .

I.1 - From the non-linear case to the linear case � With ρ ( t , x ) := ρ ∗ [1 + M p ′ ( ρ ∗ )), a ∗ r ( t , x )] ( ρ ∗ = O (1), a ∗ = . the (dimensionless) barotropic Euler system   ∂ t ρ + ∇ · ( ρ u ) = 0 , ∂ t ( ρ u ) + ∇ ( ρ u ⊗ u ) + ∇ p ( ρ ) = 0 .  M 2 is equivalent to ∂ t q + H ( q ) + L M ( q ) = 0 � �  r   q = ,   u      � �   u · ∇ r   H ( q ) = := ( u · ∇ ) q ,   ( u · ∇ ) u with      ( a ∗ + Mr ) ∇ · u            L ( q ) =  p ′ [ ρ ∗ (1 + M  .   a ∗ r )]     ∇ r  a ∗ (1 + M a ∗ r ) ◮ H = non-linear transport operator (time scale = 1); ◮ L / M = non-linear acoustic operator (time scale = M ). . On Godunov type schemes accurate at any Mach number 9 .

I.1 - From the non-linear case to the linear case • Linearization without convection: Let us define the linearization of L ( q ) with � �  . r  q = ,   u      ∇ · u       Lq = a ∗   ∇ r where a ∗ = C st 2 such that O ( a ∗ ) = 1. ◮ L / M = linear acoustic operator (time scale = M ). So, we replace the (dimensionless) barotropic Euler system ∂ t q + H ( q ) + L M ( q ) = 0 with the linear wave equation ∂ t q + L M q = 0 . (1) Let us note that (1) may be seen as a linearization of the comp. Euler system (without convection) with � � 1 + M r ( t , x ) such that p ( t , x ) := p ∗ r ( t , x ) . a ∗ In the sequel, r will be considered as a pressure perturbation . . On Godunov type schemes accurate at any Mach number 10 .

I.1 - From the non-linear case to the linear case Let us now introduce the sets . � � � � � � r ( L 2 ( T d )) 1+ d := T d r 2 dx + T d | u | 2 dx < + ∞ q := : u � equipped with the inner product � q 1 , q 2 � = T d q 1 q 2 dx and  � � q ∈ ( L 2 ( T d )) 1+ d : ∇ r = 0 and ∇ · u = 0  E = ,    � � �   E ⊥ = q ∈ ( L 2 ( T d )) 1+ d : T d rdx = 0 , ∃ φ ∈ H 1 ( T d ) , u = ∇ φ   ( T d is the torus in R d , d ∈ { 1 , 2 , 3 } ). Let us recall that: Lemma 2.1 (Hodge decomposition) E ⊕ E ⊥ = ( L 2 ( T d )) 1+ d E ⊥ E ⊥ . and In other words, any q ∈ ( L 2 ( T d )) 1+ d can be decomposed into q + q ⊥ q := P q , q ⊥ ) ∈ E × E ⊥ . q = � where ( � . On Godunov type schemes accurate at any Mach number 11 .

I.1 - From the non-linear case to the linear case • The low Mach asymptotics and the linear wave equation: Lemma 2.2 . Let q ( t , x ) be solution of the linear wave equation  ∂ t q + L   M q = 0 , (2)   q ( t = 0 , x ) = q 0 ( x ) . Thus, we have q = q 1 + q 2 with q 1 = P q 0 and q 2 = ( 1 − P ) q =: q ⊥ where q 2 is solution of (2) with the initial condition q 2 ( t = 0 , x ) = ( 1 − P ) q 0 ( x ) . Moreover, we have || q 0 − P q 0 || = O ( M ) || q − P q 0 || ( t ≥ 0) = O ( M ) . = ⇒ (3) . On Godunov type schemes accurate at any Mach number 12 .

I.2 - The perturbed linear wave equation The previous results are obtained by using the properties: . ◮ Conservation of the energy E ( t ) := � q , q � = C st . ◮ E = KerL . • We can relax these two properties with: Theorem 2.3 Let q ( t , x ) be solution of the linear PDE  ∂ t q + L   M q = 0 , (4)   q ( t = 0) = q 0 supposed to be well-posed in such a way || q || ( t ≥ 0) ≤ C || q 0 || where C does not depend on M. Then, when L is such that E ⊆ Ker L , the solution q ( t , x ) of (4) verifies || q 0 − P q 0 || = O ( M ) || q − P q 0 || ( t ≥ 0) = O ( M ) . = ⇒ . On Godunov type schemes accurate at any Mach number 13 .

I.2 - The perturbed linear wave equation Definition 2.4  ∂ t q + L .  M q = 0 ,  The solution q ( t , x ) of is said to be accurate at low Mach   q ( t = 0) = q 0 number in the incompressible regime if and only if the estimate || q 0 − P q 0 || = O ( M ) || q − P q 0 || ( t ≥ 0) = O ( M ) = ⇒ (5) is satisfied. (5) is verified (5) is NOT verified (5) is NOT verified • We deduce from Theorem 2.3 that: E ⊆ Ker L is a sufficient condition to be accurate in the sense of Definition 2.4. • The low Mach problem can be explained by replacing L with L = L + δ L where δ L = perturbation due to the spatial discretization . . On Godunov type schemes accurate at any Mach number 14 .

I.3 - The Godunov scheme applied to the linear wave equation The Godunov scheme applied to the linear wave equation is given by  � . dt r i + a ∗ d 1  M · | Γ ij | [( u i + u j ) · n ij + r i − r j ] = 0 ,    2 | Ω i |  Γ ij ⊂ ∂ Ω i � dt u i + a ∗ d 1   M · | Γ ij | [ r i + r j + κ ( u i − u j ) · n ij ] n ij = 0    2 | Ω i | Γ ij ⊂ ∂ Ω i with κ := 1. This scheme can be written in the compact form  dt q h + L κ, h d  � �  M q h = 0 , r i with q h := (6) u i   q h ( t = 0) = q 0 h Lemma 2.5 � � � � r h ∈ R 3 N Ker L κ =1 , h = q h := s.t. ∃ c , ∀ i : r i = c and ( u i − u j ) · n ij = 0 u h   � �   � | Γ ij | u i + u j r h ∈ R 3 N Ker L κ =0 , h =  q h := s.t. ∃ c , ∀ i : r i = c and · n ij = 0  . u h 2 Γ ij ⊂ ∂ Ω i � � u i + u j Do we have E h ⊆ Ker L κ, h ??? Let us note that | Γ ij | · n ij ≃ Ω i ∇ · u dx . 2 Γ ij ⊂ ∂ Ω i . On Godunov type schemes accurate at any Mach number 15 .