Analysis of Asymptotic Preserving schemes with the modified equation - PowerPoint PPT Presentation

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno Despr´ es (LJLL-UPMC) collaboration with C. Buet (CEA) et E. Franck (PhD-CEA-LJLL) Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 1 / 25 Bruno

1D telegraph equation Model problem (stiff source terms, radiation , friction, . . .)  ∂ t u ε + 1   ε∂ x v ε = 0 ,   Model problem   ∂ t v ε + 1 ε∂ x u ε + σ  Modified  ε 2 v ε = 0 . equations Main result σ > 0 is a given coefficient. The small parameter is 0 < ε ≤ 1. Schemes Numerical Hilbert expansion with respect to the small parameter ε results u ε = u 0 + · · · , v ε = v 0 + ε v 1 + · · · yields v 0 = 0 and ∂ x u 0 + σ v 1 = 0. It justifies the limit diffusion equation ∂ t u 0 − 1 σ∂ xx u 0 = 0 . Jin-Levermore , Numerical methods for hyperbolic conservation laws with stiff relaxation terms, JCP 1996. Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 2 / 25 Bruno

Standard discretization The usual F.V scheme (Riemann) writes  u j − u j + v j +1 − v j − 1 − u j +1 − 2 u j + u j − 1   = 0 ,  Model ∆ t 2 ε ∆ x 2 ε ∆ x problem  v j − v j + u j +1 − u j − 1 − v j +1 − 2 v j + v j − 1 + σ Modified   ε 2 u j = 0 . equations ∆ t 2 ε ∆ x 2 ε ∆ x Main result Schemes Hilbert expansion ( u j = u 0 j + · · · and v j = v 0 j + ε v 1 j + · · · ) Numerical yields results u 0 j +1 − u 0 j − 1 v 0 + σ v 1 j = 0 , j = 0 . 2∆ x The limit scheme is u j 0 − u 0 u 0 j +2 − 2 u 0 j + u 0 u 0 j +1 − 2 u 0 j + u 0 − 1 − ∆ x j − 2 j − 1 j = 0 . ∆ x 2 ∆ x 2 ∆ t 2 ε σ Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 3 / 25 Bruno

Summary • The limit scheme is consistent with � 1 � σ + ∆ x ∂ t u − ∂ xx u = 0 . 2 ε Model problem Modified This is not correct in the regime ∆ x ≥ ε ! ! equations Main result • The main objective of A.P. methods is to modify the basic Schemes scheme. Numerical results • There exists different approaches : Jin-Levermore, Gosse-Toscani, . . ., Jin-Pareschi, . . .. The structure of these methods may seem strange , in particular for the most efficient methods. • First goal of this presentation : can we understand a priori the necessary modifications, starting from PDEs ? Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 4 / 25 Bruno

Modified equations • Start from the modified equation  Model ∂ t u ε,α + 1 problem   ε ( ∂ x v ε,α − α∂ xx u ε,α ) = 0 , Modified equations  ∂ t v ε,α + 1 ε ( ∂ x u ε,α − α∂ xx u ε,α ) = − σ  ε 2 v ε,α , Main result Schemes Numerical with two small parameters : ε and α ≈ ∆ x 2 . results • Reminder : Hilbert expansion with respect to ε ( α > 0 fixed) is not correct � 1 � σ + α ∂ t u − ∂ xx u = 0 ε Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 5 / 25 Bruno

Modified equation of the first kind • Introduce the Magic coefficient M = ε + σα ∈ ]0 , 1] together ε with  Model ∂ t u ε,α + M problem   ε ( ∂ x v ε,α − α∂ xx u ε,α ) = 0 , Modified equations  ∂ t v ε,α + 1 ε ( ∂ x u ε,α − α∂ xx v ε,α ) = − σ  Main result ε 2 v ε,α , Schemes Numerical results • Hilbert expansion with respect to ε ( α > 0 fixed) yields � 1 � σ + α ∂ t u 0 ,α − M ∂ xx u 0 ,α = 0 . ε � 1 � ε + σα = 1 This is correct now since : M σ + α = ε σ . ε + ασ ε σε Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 6 / 25 Bruno

Modified equation of the second kind • The symetry in the equations is better using the Magic coefficient in both equations  u α,ε + M  Model  ∂ t � ε ( ∂ x � v α,ε − α∂ xx � u α,ε ) = 0 , problem Modified  v α,ε + M v α,ε ) = − σ M  equations ∂ t � ε ( ∂ x � u α,ε − α∂ xx � ε 2 � v α,ε , Main result Schemes Numerical results • Once again, Hilbert expansion is correct u α, 0 − 1 ∂ t � σ∂ xx � u α, 0 = 0 . • The identity energy is correct in the sense � � u α,ε ( t ) 2 + � u α,ε (0) 2 + � v α,ε ( t ) 2 v α,ε (0) 2 � � dx ≤ dx . 2 2 R R Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 7 / 25 Bruno

Main result An extension of a result proved for numerical methods in Design of asymptotic preserving finite volume schemes for Model the hyperbolic heat equation on unstructured meshes , problem Modified Numer. Math., 2012, with C. Buet and E. Franck yields equations Main result Theor : Consider well prepared data Schemes Numerical v 0 = − ε u 0 ∈ H 3 ( R ) , σ∂ x u 0 + ε 2 v 2 v 2 ∈ H 1 ( R ) . results There exists C > 0 ind´ ependent of ε such that ( t ≤ T ) � u ε ( t ) − � u α,ε ( t ) � L 2 ( R ) + � v ε ( t ) − � v α,ε ( t ) � L 2 ( R ) ≤ C min( α, ε ) . Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 8 / 25 Bruno

Sketch of the proof Define e = u ε − � u α,ε and f = v ε − � v α,ε One has  ∂ t e + M   ε ( ∂ x f − α∂ xx e ) = r , Model problem  ∂ t f + M ε ( ∂ x f − α∂ xx f ) + σ M  ε 2 f = s , Modified equations Main result with Schemes r = ∂ t u ε + M ε ( ∂ x v ε − α∂ xx u ε ) = (1 − M ) ∂ t u ε − M α ε ∂ xx u ε , Numerical results s = ∂ t v ε + M ε ( ∂ x u ε − α∂ xx v ε ) + σ M ε 2 v ε = (1 − M ) ∂ t v ε − M α ε ∂ xx v ε . Main intermediate result � r � L ∞ ([0 , T ]: L 2 ( R )) + � s � L 2 ([0 , T ]: L 2 ( R )) ≤ C min( α, ε ) which proves the claim. Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 9 / 25 Bruno

Philosophy Model An A.P. scheme with good properties introduces the problem Magic coefficient M at the right place. Modified equations Main result Schemes The design principle of the A.P. scheme is less important. Numerical results It may be a full Riemann solver, a simplified one, coming from W.B. (well balanced) techniques, a direct finite difference scheme, . . . Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 10 / 25 Bruno

Jin-Levermore Riemann invariants v ± u are modified � v n Model j + u n = v n 2 + u n + σ ∆ x 2 ε v n 2 , problem j j + 1 j + 1 j + 1 2 Modified v n j +1 − v n = v n 2 − u n + σ ∆ x 2 ε v n 2 . equations j +1 j + 1 j + 1 j + 1 2 Main result The solution of this linear system is Schemes Numerical  � � results  2 = M v n v n j + v n j +1 + u n j − u n , j + 1 j +1 2 � �  2 = 1 u n u n j + u n j +1 + v n j − v n . j + 1 2 j +1 where M = 2 σ is the correct value. ε ε + ∆ x Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 11 / 25 Bruno

Final scheme 2 = Mv Classique Model Since v n , one obtains the Jin-Levermore problem j + 1 j + 1 2 Modified scheme equations  Main result u j − u j + M v j +1 − v j − 1 − M u j +1 − 2 u j + u j − 1   = 0 , Schemes ∆ t 2 ε ∆ x 2 ε ∆ x v j − v j + u j +1 − u j − 1 − v j +1 − 2 v j + v j − 1 Numerical + σ   results ε 2 v j = 0 . ∆ t 2 ε ∆ x 2 ε ∆ x It corresponds to the modified equation of the first kind. Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 12 / 25 Bruno

Gosse-Toscani scheme Moreover replace the source term σ ε 2 v j by Model problem 2 + v j + 1 v j − 1 σ Modified 2 . equations ε 2 2 Main result One obtains exactly Schemes  Numerical u j − u j + M v j +1 − v j − 1 − M u j +1 − 2 u j + u j − 1  results  = 0 , ∆ t 2 ε ∆ x 2 ε ∆ x v j − v j + M u j +1 − u j − 1 − M v j +1 − 2 v j + v j − 1 + M σ   ε 2 v j = 0 , ∆ t 2 ε ∆ x 2 ε ∆ x It corresponds to the modified equation of the second kind. Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 13 / 25 Bruno

MultiD : u ε ∈ R , v ε ∈ R 2 Solve : ∂ t u ε + 1 ∂ t v ε + 1 ε ∇ u ε = − σ ε ∇ · v ε = 0 , ε 2 v ε . Model problem 2 2 Modified 1.5 1.5 equations Main result 1 1 Schemes 0.5 0.5 Numerical results 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 14 / 25 Bruno

Jin-Levermore on edges Model x r problem Cell Ω k Modified x r + 1 equations x k l jk Main result n jk Schemes x j Numerical Cell Ω j results x r − 1 First problem : x j x k is not parallel to normal vector n jk . Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes p. 15 / 25 Bruno