An admissibility and asymptotic-preserving scheme for systems of - PowerPoint PPT Presentation

An admissibility and asymptotic-preserving scheme for systems of conservation laws with source terms on 2D unstructured meshes F. Blachère 1 R. Turpault 1 1 Laboratoire de Mathématiques Jean Leray, Université de Nantes SHARK-FV14, 1 st May 2014

Outline General context and examples 1 State-of-the-art 2 Development of a new asymptotic preserving FV scheme 3 Conclusion and perspectives 4 F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 2 / 31

Outline General context and examples 1 State-of-the-art 2 Development of a new asymptotic preserving FV scheme 3 Conclusion and perspectives 4 F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 3 / 31

Problematic Hyperbolic systems of conservation laws with source terms: ∂ t U + div ( F ( U )) = γ ( U )( R ( U ) − U ) (1) A : set of admissible states, U ∈ A ⊂ R N , F : flux, γ > 0: controls the stiffness, R : A → A : smooth function with some compatibility conditions developed by C. Berthon – P.G. Le Floch – R. Turpault [3]. F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 4 / 31

Problematic Hyperbolic systems of conservation laws with source terms: ∂ t U + div ( F ( U )) = γ ( U )( R ( U ) − U ) (1) Under compatibility conditions on R , when γ t → ∞ , (1) degenerates into a smaller parabolic system: � � ∂ t u − div M ( u ) ∇ u = 0 (2) u ∈ R n , linked to U , M : positive and definite matrix. F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 5 / 31

Examples: Telegraph equations � ∂ t v + a ∂ x v = σ ( w − v ) σ ( v − w ) , a , σ > 0 ∂ t w − a ∂ x w = Formalism of (1) U = ( v , w ) T F ( U ) = ( av , − aw ) T R ( U ) = ( w , v ) T γ ( U ) = σ Limit diffusion equation: heat equation on ( v + w ) � a 2 � ∂ t ( v + w ) − ∂ x 2 σ∂ x ( v + w ) = 0 F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 6 / 31

Examples: isentropic Euler with friction  ∂ t ρ + ∂ x ρ u + ∂ y ρ v = 0  ∂ t ρ u + ∂ x ( ρ u 2 + p ( ρ )) + ∂ y ρ uv , with: p ′ ( ρ ) > 0 , κ > 0 = − κρ u ∂ t ρ v + ∂ x ρ uv + ∂ y ( ρ v 2 + p ( ρ )) = − κρ v  A = { ( ρ, ρ u , ρ v ) T ∈ R 3 /ρ > 0 } Formalism of (1) � ρ u , ρ u 2 + p � T , ρ uv U = ( ρ, ρ u , ρ v ) T F ( U ) = , ρ v 2 + p ρ v , ρ uv R ( U ) = ( ρ, 0 , 0 ) T γ ( U ) = κ Limit diffusion equation � 1 � ∂ t ρ − div κ ∇ p ( ρ ) = 0 F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 7 / 31

Examples: M1 model for radiative transfer c σ e aT 4 − c σ a E  ∂ t E + ∂ x F x + ∂ y F y =  − c σ f F x  ∂ t F x + c 2 ∂ x P xx ( E , F ) + c 2 ∂ y P xy ( E , F ) =  − c σ f F y ∂ t F y + c 2 ∂ x P yx ( E , F ) + c 2 ∂ y P yy ( E , F ) =   c σ a E − c σ e aT 4 ρ C v ∂ t T =  σ = σ ( E , F x , F y , T ) � A = { ( E , F x , F y , T ) ∈ R 4 / E > 0 , T > 0 , F 2 x + F 2 y < cE } Formalism of (1): � F x , � T c 2 P xx , c 2 P yx , 0 U = ( E , F x , F y , T ) T F ( U ) = c 2 P xy , c 2 P yy F y , , 0 R ( U ) γ ( U ) = c σ m ( U ) Limit diffusion equation: equilibrium diffusion equation � c � ∂ t ( ρ C v T + aT 4 ) − div 3 σ r ∇ ( aT 4 ) = 0 F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 8 / 31

Other examples Euler coupled with the M1 model − → diffusion system F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 9 / 31

Other examples Euler coupled with the M1 model − → diffusion system Shallow water with friction � ∂ t h + ∂ x hv = 0 − κ ( h ) 2 ghv | hv | , with: κ ( h ) = κ 0 ∂ t hv + ∂ x ( hv 2 + gh 2 h η 2 ) = Limit diffusion equation: non linear parabolic equation � √ � h ∂ x h ∂ t h − ∂ x = 0 κ ( h ) � | ∂ x h | F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 9 / 31

Aim of an AP scheme Diffusion system: Model: γ t → ∞ � � ∂ t U + div ( F ( U )) = γ ( U )( R ( U ) − U ) ∂ t u − div M ( u ) ∇ u = 0 consistent: not consistent: ∆ t , ∆ x → 0 ∆ t , ∆ x → 0 Numerical scheme Limit scheme γ t → ∞ F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 10 / 31

Aim of an AP scheme Diffusion system: Model: γ t → ∞ � � ∂ t U + div ( F ( U )) = γ ( U )( R ( U ) − U ) ∂ t u − div M ( u ) ∇ u = 0 consistent: not consistent: ∆ t , ∆ x → 0 ∆ t , ∆ x → 0 Numerical scheme Limit scheme γ t → ∞ F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 10 / 31

Example of a non AP scheme on Euler with friction in 1D ∂ t U + ∂ x ( F ( U )) = γ ( U )( R ( U ) − U ) ( ρ u , ρ u 2 + p ) T ( ρ, ρ u ) T = F ( U ) = U ( ρ, 0 ) T γ ( U ) = κ R ( U ) = F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 11 / 31

Example of a non AP scheme on Euler with friction in 1D ∂ t U + ∂ x ( F ( U )) = γ ( U )( R ( U ) − U ) ( ρ u , ρ u 2 + p ) T ( ρ, ρ u ) T = F ( U ) = U ( ρ, 0 ) T γ ( U ) = κ R ( U ) = U n + 1 − U n = − 1 i i + γ ( U n i )( R ( U n i ) − U n � � F i + 1 / 2 − F i − 1 / 2 i ) ∆ t ∆ x F i + 1 / 2 : Rusanov flux F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 11 / 31

Example of a non AP scheme on Euler with friction in 1D ∂ t U + ∂ x ( F ( U )) = γ ( U )( R ( U ) − U ) ( ρ u , ρ u 2 + p ) T ( ρ, ρ u ) T = F ( U ) = U ( ρ, 0 ) T γ ( U ) = κ R ( U ) = U n + 1 − U n = − 1 i i + γ ( U n i )( R ( U n i ) − U n � � F i + 1 / 2 − F i − 1 / 2 i ) ∆ t ∆ x F i + 1 / 2 : Rusanov flux Limit ρ n + 1 − ρ n 1 i i b i + 1 / 2 ∆ x ( ρ n i + 1 − ρ n i ) − b i − 1 / 2 ∆ x ( ρ n i − ρ n � � = i − 1 ) 2 ∆ x 2 ∆ t F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 11 / 31

Outline General context and examples 1 State-of-the-art 2 Development of a new asymptotic preserving FV scheme 3 Conclusion and perspectives 4 F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 12 / 31

AP in 1D 1 controls the numerical diffusion: telegraph equations: L. Gosse – G. Toscani [11], M1 model: C. Buet – B. Desprès [7], C. Buet – S. Cordier [6] , C. Berthon – P. Charrier – B. Dubroca [2], . . . F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 13 / 31

AP in 1D 1 controls the numerical diffusion: telegraph equations: L. Gosse – G. Toscani [11], M1 model: C. Buet – B. Desprès [7], C. Buet – S. Cordier [6] , C. Berthon – P. Charrier – B. Dubroca [2], . . . 2 ideas of hydrostatic reconstruction used in ‘well-balanced’ scheme: used to have AP properties Euler with friction: F. Bouchut – H. Ounaissa – B. Perthame [5] F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 13 / 31

AP in 1D 1 controls the numerical diffusion: telegraph equations: L. Gosse – G. Toscani [11], M1 model: C. Buet – B. Desprès [7], C. Buet – S. Cordier [6] , C. Berthon – P. Charrier – B. Dubroca [2], . . . 2 ideas of hydrostatic reconstruction used in ‘well-balanced’ scheme: used to have AP properties Euler with friction: F. Bouchut – H. Ounaissa – B. Perthame [5] 3 using convergence speed and finite differences: D. Aregba-Driolet – M. Briani – R. Natalini [1] F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 13 / 31

AP in 1D 1 controls the numerical diffusion: telegraph equations: L. Gosse – G. Toscani [11], M1 model: C. Buet – B. Desprès [7], C. Buet – S. Cordier [6] , C. Berthon – P. Charrier – B. Dubroca [2], . . . 2 ideas of hydrostatic reconstruction used in ‘well-balanced’ scheme: used to have AP properties Euler with friction: F. Bouchut – H. Ounaissa – B. Perthame [5] 3 using convergence speed and finite differences: D. Aregba-Driolet – M. Briani – R. Natalini [1] 4 generalization of L. Gosse – G. Toscani: C. Berthon – P. Le Floch – R. Turpault [3] F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 13 / 31

AP in 2D cartesian and admissible meshes = ⇒ 1D F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 14 / 31

AP in 2D cartesian and admissible meshes = ⇒ 1D unstructured meshes: MPFA based scheme: 1 C. Buet – B. Desprès – E. Frank [8] using the diamond scheme (Y. Coudière – J.P. Vila – P. Villedieu [9]) 2 for the limit scheme: C. Berthon – G. Moebs - C. Sarazin-Desbois – R. Turpault [4] F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 14 / 31

Outline General context and examples 1 State-of-the-art 2 Development of a new asymptotic preserving FV scheme 3 Conclusion and perspectives 4 F. Blachère, R. Turpault (Nantes) AP schemes on 2D unstructured meshes SHARK-FV14, 01/05/14 15 / 31