Admissibility and asymptotic-preserving scheme . Blachre 1 , R. - PowerPoint PPT Presentation

Admissibility and asymptotic-preserving scheme . Blachère 1 , R. Turpault 2 F 1 Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes, 2 Institut de Mathématiques de Bordeaux (IMB), Bordeaux-INP GdR EGRIN, 04/06/2015, Piriac-sur-Mer

Outline General context and examples 1 Development of a new asymptotic preserving FV scheme 2 Conclusion and perspectives 3 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28

Outline General context and examples 1 Development of a new asymptotic preserving FV scheme 2 Conclusion and perspectives 3 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28

Problematic Hyperbolic systems of conservation laws with source terms: ∂ t W + div ( F ( W )) = γ ( W )( R ( W ) − W ) (1) A : set of admissible states, W ∈ A ⊂ R N , F : physical flux, γ > 0: controls the stiffness, R : A → A : smooth function with some compatibility conditions (cf. [Berthon, LeFloch, and Turpault, 2013]). F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 2/28

Problematic Hyperbolic systems of conservation laws with source terms: ∂ t W + div ( F ( W )) = γ ( W )( R ( W ) − W ) (1) Under compatibility conditions on R , when γ t → ∞ , (1) degenerates into a diffusion equation: � � ∂ t w − div D ( w ) ∇ w = 0 (2) w ∈ R , linked to W , D : positive and definite matrix, or positive function. F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 3/28

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 W = ( ρ, ρ u , ρ v ) T F ( W ) = , ρ v 2 + p ρ v , ρ uv R ( W ) = ( ρ, 0 , 0 ) T γ ( W ) = κ Limit diffusion equation � 1 � ∂ t ρ − div κ ∇ p ( ρ ) = 0 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 4/28

Examples: shallow water equations (linear friction)  ∂ t h + ∂ x hu + ∂ y hv = 0  ∂ t hu + ∂ x ( hu 2 + gh 2 / 2 ) + ∂ y huv − κ = h η hu , with: κ > 0 ∂ t hv + ∂ x huv + ∂ y ( hv 2 + gh 2 / 2 ) − κ = h η hv  A = { ( h , hu , hv ) T ∈ R 3 / h > 0 } Formalism of (1) � hu , hu 2 + p � T , huv W = ( h , hu , hv ) T F ( W ) = , hv 2 + p hv , huv R ( W ) = ( h , 0 , 0 ) T γ ( W ) = κ h η Limit diffusion equation � h η κ ∇ gh 2 � ∂ t h − div = 0 2 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 5/28

Examples: shallow water equations (Manning friction)  ∂ t h + ∂ x hu + ∂ y hv = 0  ∂ t hu + ∂ x ( hu 2 + gh 2 / 2 ) + ∂ y huv − κ = h η � h u � hu ∂ t hv + ∂ x huv + ∂ y ( hv 2 + gh 2 / 2 ) − κ = h η � h u � hv  A = { ( h , hu , hv ) T ∈ R 3 / h > 0 } Formalism of (1) � hu , hu 2 + p � T , huv W = ( h , hu , hv ) T F ( W ) = , hv 2 + p hv , huv R ( W ) = ( h , 0 , 0 ) T γ ( W ) = κ h η � h u � Limit equation √ �� � gh η h ∂ t h − div ∇ h = 0 � κ �∇ h � F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 6/28

Aim of an AP scheme Diffusion equation: Model: γ t → ∞ � � ∂ t w − div D ( w ) ∇ w = 0 ∂ t W + div ( F ( W )) = γ ( W )( R ( W ) − W ) consistent: consistent? ∆ t , ∆ x → 0 Numerical scheme Limit scheme γ t → ∞ F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 7/28

Example of a non AP scheme in 1D ∂ t W + ∂ x ( F ( W )) = γ ( W )( R ( W ) − W ) (1) ( ρ u , ρ u 2 + p ) T ( ρ, ρ u ) T = F ( W ) = W ( ρ, 0 ) T γ ( W ) = κ R ( W ) = x i − 1 / 2 x i +1 / 2 x i +1 x i − 1 x i W n + 1 − W n = − 1 i i + γ ( W n i )( R ( W n i ) − W n � � F i + 1 / 2 − F i − 1 / 2 i ) ∆ t ∆ x 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 ) ∆ t 2 ∆ x 2 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 8/28

State-of-the-art for AP schemes in 1D 1 control of numerical diffusion: telegraph equations: [Gosse and Toscani, 2002], M1 model: [Buet and Després, 2006], [Buet and Cordier, 2007], [Berthon, Charrier, and Dubroca, 2007], . . . Euler with gravity and friction: [Chalons, Coquel, Godlewski, Raviart, and Seguin, 2010], 2 ideas of hydrostatic reconstruction used in ‘well-balanced’ scheme used to have AP properties: Euler with friction: [Bouchut, Ounaissa, and Perthame, 2007], 3 using convergence speed and finite differences: [Aregba-Driollet, Briani, and Natalini, 2012], 4 generalization of Gosse and Toscani: [Berthon and Turpault, 2011], [Berthon, LeFloch, and Turpault, 2013]. F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 9/28

State-of-the-art for AP schemes in 2D Cartesian and admissible meshes = ⇒ 1D unstructured meshes: MPFA based scheme: 1 [Buet, Després, and Franck, 2012], using the diamond scheme (Coudière, Vila, and Villedieu) for the limit 2 scheme: [Berthon, Moebs, Sarazin-Desbois, and Turpault, 2015], SW with Manning-type friction: 3 [Duran, Marche, Turpault, and Berthon, 2015]. F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 10/28

Outline General context and examples 1 Development of a new asymptotic preserving FV scheme 2 Conclusion and perspectives 3 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28

Notations M is an unstructured mesh n K,i 1 constituted of polygonal cells K , x L 2 x L 1 x K is the center of the cell K , i 1 L is the neighbour of K by the x L 5 i 2 interface i , i = K ∩ L , x K i 5 E K are the interfaces of K , x L 3 | e i | is length of the interface i , i 3 i 4 | K | is the area of the cell K , x L 4 n K , i is the normal vector to the x J 2 interface i . F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 11/28

Aim of the development for any 2D unstructured meshes, for any system of conservation laws which could be written as (1), under a ‘hyperbolic’ CFL: stability, preservation of A , preservation of the asymptotic behaviour, � ∆ t � ≤ 1 max b K , i 2 . ∆ x K ∈M i ∈E K F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 12/28

Outline General context and examples 1 Development of a new asymptotic preserving FV scheme 2 Choice of a limit scheme Hyperbolic part Numerical results for the hyperbolic part Scheme for the complete system Results for the complete system Conclusion and perspectives 3 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28

Aim of an AP scheme Diffusion equation: Model: γ t → ∞ � � ∂ t w − div D ( w ) ∇ w = 0 ∂ t W + div ( F ( W )) = γ ( W )( R ( W ) − W ) consistent: consistent? ∆ t , ∆ x → 0 Numerical scheme Limit scheme γ t → ∞ F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 13/28

Choice of the limit scheme FV scheme to discretize diffusion equations: ∂ t w − div ( D ( w ) ∇ w ) = 0 (2) Choice: scheme developed by Droniou and Le Potier conservative and consistent, preserves A , nonlinear. � ν J ( D · ∇ i w K ) · n K , i = K , i ( w )( w J − w K ) J ∈ S K , i S K , i the set of points used for the reconstruction on edges i of cell K ν J K , i ( w ) ≥ 0 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 14/28

Presentation of the DLP scheme x A M K,i x L n K , i n L , i x K M L,i i x B J ∈ S K , i ω J J ∈ S K , i ω J � � M K , i = K , i X J w M K , i = K , i w J J ∈ S L , i ω J J ∈ S L , i ω J M L , i = � L , i X J w M L , i = � L , i w J F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 15/28

Presentation of the DLP scheme Two approximations w MK , i − w K ∇ i w K · n K , i = | KM K , i | w ML , i − w L ∇ i w L · n L , i = | LM L , i | Convex combination: γ K , i + γ L , i = 1, γ K , i ≥ 0, γ L , i ≥ 0 ∇ i w K · n K , i = γ K , i ( w ) ∇ i w K · n K , i + γ L , i ( w ) ∇ i u L · n L , i J ∈ S K , i ν J K , i ( w )( w J − w K ) , with : ν J � = K , i ( w ) ≥ 0 Properties of the DLP scheme consistent with the diffusion equation on any mesh, satisfy the maximum principle. F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 16/28

Outline General context and examples 1 Development of a new asymptotic preserving FV scheme 2 Choice of a limit scheme Hyperbolic part Numerical results for the hyperbolic part Scheme for the complete system Results for the complete system Conclusion and perspectives 3 F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 1/28

Aim of an AP scheme Diffusion equation: Model: γ t → ∞ � � ∂ t w − div D ( w ) ∇ w = 0 ∂ t W + div ( F ( W )) = γ ( W )( R ( W ) − W ) consistent: consistent? ∆ t , ∆ x → 0 Numerical scheme Limit scheme γ t → ∞ F. Blachère (LMJL) GdR EGRIN, 04/06/2015, Piriac-sur-Mer 17/28