Nodal finite volumes for hyperbolic systems with source terms on - PowerPoint PPT Presentation

Nodal finite volumes for hyperbolic systems with source terms on unstructured meshes E. Franck 1 , C. Buet 2 , B. Despr´ es 3 T.Leroy 2 , 3 , L. Mendoza 4 July 31, 2015 1 INRIA Nancy Grand-Est and IRMA Strasbourg, TONUS team, France 2 CEA DAM, Arpajon, France 3 LJLL, UPMC, France 4 IPP, Garching bei M¨ unchen, Germany 1 / 33 E. Franck FV nodal schemes 1/33

Outline Mathematical and physical context AP scheme for the P 1 model Extension to the Euler model 2 / 33 E. Franck FV nodal schemes 2/33

Mathematic and physical context 3 / 33 E. Franck FV nodal schemes 3/33

Stiff hyperbolic systems � Stiff hyperbolic system with source terms : ∂ t U + 1 ε ∂ x F ( U ) + 1 ε ∂ y G ( U ) = 1 ε S ( U ) − σ ε 2 R ( U ) , U ∈ R n with ε ∈ ] 0, 1 ] et σ > 0. � Subset of solutions given by the balance between the source terms and the convective part: � Diffusion solutions for ε → 0 and S ( U ) = 0: ∂ t V − div ( K ( ∇ V , σ )) = 0, V ∈ Ker R . � Steady states for σ = 0 et ε → 0 : ∂ x F ( U ) + ∂ y G ( U ) = S ( U ) . � Applications: biology, neutron transport, fluid mechanics, plasma physics, Radiative hydrodynamic for inertial fusion (hydrodynamic + linear transport of photon). 4 / 33 E. Franck FV nodal schemes 4/33

Well-Balanced schemes � Discretization of physical steady states is important (Lack at rest for Shallow water equations, hydrostatic equilibrium for astrophysical flows ..) � Classical scheme : the physical steady states or a good discretization of the steady states are not the equilibriums of the scheme. � Consequence : Spurious numerical velocities larger than physical velocities for nearly or exact uniform flows. WB scheme: definitions � Exact Well-Balanced scheme : is a scheme exact for continuous steady-states. � Well-Balanced scheme : is a scheme exact for discrete steady-states at the interfaces. � For shallow water model : in general the schemes are exact WB schemes. � For Euler model : in general the schemes are WB schemes. 5 / 33 E. Franck FV nodal schemes 5/33

Sch´ emas ”Asymptotic preserving” � P 1 model:  ∂ t E + 1  � 1 �  ε ∂ x F = 0, − → ∂ t E − ∂ x σ ∂ x E = 0. ∂ t F + 1 ε ∂ x E = − σ   ε 2 F , � Consistency of Godunov-type ε → 0 P ε P 0 schemes: O ( ∆ x h h ε + ∆ t ) . � CFL condition: ∆ t ( 1 ∆ x ε + σ ε 2 ) ≤ 1. h → 0 h → 0 � Consistency of AP schemes: O ( ∆ x + ∆ t ) . P ε P 0 � CFL condition: � � ε → 0 1 ∆ t ≤ 1. ∆ x ε + ∆ x 2 Figure: AP diagram σ � AP vs non AP schemes: Important reduction of CPU cost. � AP schemes are obtained plugging the source term into the fluxes (WB technic). 6 / 33 E. Franck FV nodal schemes 6/33

AP Godunov schemes � Jin-Levermore scheme � Principle : plug the balance law ∂ x E = − σ ε F + O ( ε 2 ) in the fluxes. 7 / 33 E. Franck FV nodal schemes 7/33

AP Godunov schemes � Jin-Levermore scheme � Principle : plug the balance law ∂ x E = − σ ε F + O ( ε 2 ) in the fluxes. we write the relations � E ( x j ) = E ( x j + 1 2 ) + ( x j − x j + 1 2 ) ∂ x E ( x j + 1 2 ) , E ( x j + 1 ) = E ( x j + 1 2 ) + ( x j + 1 − x j + 1 2 ) ∂ x E ( x j + 1 2 ) . 7 / 33 E. Franck FV nodal schemes 7/33

AP Godunov schemes � Jin-Levermore scheme � Principle : plug the balance law ∂ x E = − σ ε F + O ( ε 2 ) in the fluxes. we write the relations � 2 ) σ E ( x j ) = E ( x j + 1 2 ) − ( x j − x j + 1 ε F ( x j + 1 2 ) , 2 ) σ E ( x j + 1 ) = E ( x j + 1 2 ) − ( x j + 1 − x j + 1 ε F ( x j + 1 2 ) . 7 / 33 E. Franck FV nodal schemes 7/33

AP Godunov schemes � Jin-Levermore scheme � Principle : plug the balance law ∂ x E = − σ ε F + O ( ε 2 ) in the fluxes. we write the relations � 2 ) σ E ( x j ) = E ( x j + 1 2 ) − ( x j − x j + 1 ε F ( x j + 1 2 ) , 2 ) σ E ( x j + 1 ) = E ( x j + 1 2 ) − ( x j + 1 − x j + 1 ε F ( x j + 1 2 ) . We couple these relations with the fluxes � F j + E j = F j + 1 2 + E j + 1 2 , F j + 1 − E j + 1 = F j + 1 2 − E j + 1 2 . 7 / 33 E. Franck FV nodal schemes 7/33

AP Godunov schemes � Jin-Levermore scheme � Principle : plug the balance law ∂ x E = − σ ε F + O ( ε 2 ) in the fluxes. we write the relations � 2 ) σ E ( x j ) = E ( x j + 1 2 ) − ( x j − x j + 1 ε F ( x j + 1 2 ) , 2 ) σ E ( x j + 1 ) = E ( x j + 1 2 ) − ( x j + 1 − x j + 1 ε F ( x j + 1 2 ) . � 2 + σ ∆ x F j + E j = F j + 1 2 + E j + 1 2 ε F j + 1 2 , 2 + σ ∆ x F j + 1 − E j + 1 = F j + 1 2 − E j + 1 2 . 2 ε F j + 1 7 / 33 E. Franck FV nodal schemes 7/33

AP Godunov schemes � Jin-Levermore scheme � Principle : plug the balance law ∂ x E = − σ ε F + O ( ε 2 ) in the fluxes. Jin Levermore scheme:  E n + 1 − E n F n j + 1 − F n E n j + 1 − 2 E n j + E n  j − 1 j − 1 j j + M − M = 0, ∆ t 2 ε ∆ x 2 ε ∆ x F n + 1 − F n E n j + 1 − E n F n j + 1 − 2 F n j + F n  j j j − 1 j − 1 + σ ε 2 F n + − j = 0, ∆ t 2 ε ∆ x 2 ε ∆ x 2 ε with M = 2 ε + σ ∆ x . 7 / 33 E. Franck FV nodal schemes 7/33

AP Godunov schemes � Jin-Levermore scheme � Principle : plug the balance law ∂ x E = − σ ε F + O ( ε 2 ) in the fluxes. Gosse-Toscani scheme:  E n + 1 − E n F n j + 1 − F n E n j + 1 − 2 E n j + E n  j − 1 j − 1 j j + M − M = 0, ∆ t 2 ε ∆ x 2 ε ∆ x F n + 1 − F n E n j + 1 − E n F n j + 1 − 2 F n j + F n  j j j − 1 j − 1 + M σ ε 2 F n + M − M j = 0, ∆ t 2 ε ∆ x 2 ε ∆ x 2 ε avec M = 2 ε + σ ∆ x . � consistency error for the � Principle of GT scheme : JL-scheme with the source term Jin-Levermore scheme: 1 2 ( F j + 1 2 + F j − 1 2 ) gives the � first equation: � � ∆ x 2 + ε ∆ x + ∆ t Gosse-Toscani scheme. O , � second equation: � Consistency error of the � � ∆ x 2 Gosse-Toscani scheme: O + ∆ x + ∆ t . ε O ( ∆ x + ∆ t ) . � � 1 � 1 � ≤ 1. � Explicit CFL: ∆ t ∆ x ε + σ ≤ 1. ε 2 � Explicit CFL: ∆ t ∆ x ε � 1 � ≤ 1. � Semi-implicit CFL: ∆ t � Semi-implicit CFL : ∆ x ε � � 1 ∆ t ≤ 1. ∆ x ε + ∆ x 2 7 / 33 E. Franck FV nodal schemes 7/33

Numerical example � Validation test for AP scheme : the data are E ( 0, x ) = G ( x ) with G ( x ) a Gaussian F ( 0, x ) = 0 and σ = 1, ε = 0.001. Ap scheme Godunov scheme L 1 error Scheme CPU time Godunov, 10000 cells 0.0366 1485m4.26s Godunov, 500 cells 0.445 0m24.317s AP, 500 cells 0.0001 0m15.22s AP, 50 cells 0.0065 0m0.054s 8 / 33 E. Franck FV nodal schemes 8/33

Non complete state of art � S. Jin, D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms , (1996). � C. Berthon, R. Turpault, Asymptotic preserving HLL schemes , (2012). � L. Gosse, G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations , (2002). � C. Berthon, P. Charrier and B. Dubroca, An HLLC scheme to solve the M 1 model of radiative transfer in two space dimensions , (2007). � C. Chalons, M. Girardin, S. Kokh, Large time step asymptotic preserving numerical schemes for the gas dynamics equations with source terms , (2013). � C. Chalons, F. Coquel, E. Godlewski, P-A. Raviart, N. Seguin, Godunov-type schemes for hyperbolic systems with parameter dependent source , (2010). � R. Natalini and M. Ribot, An asymptotic high order mass-preserving scheme for a hyperbolic model of chemotaxis , (2012). � M. Zenk, C. Berthon et C. Klingenberg, A well-balanced scheme for the Euler equations with a gravitational potential , (2014). � J. Greenberg, A. Y. Leroux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations , (1996). � R. Kappeli, S. Mishra, Well-balanced schemes for the Euler equations with gravitation , (2013). 9 / 33 E. Franck FV nodal schemes 9/33

Why unstructured meshes ? � Applications : coupling between radiation and hydrodynamic 1 1 � In some hydrodynamic codes : Lagrangian or ALE scheme cell-centered for multi-material problems. � Example of meshes obtained using a ALE code. � Aim : Design and analyze AP cell-centered for linear transport on general meshes. 0 0 −1.5×10 −18 −1.5×10 −18 1.0×10 0 1.0×10 0 10 / 33 E. Franck FV nodal schemes 10/33

Sch´ emas ”Asymptotic preserving” 2D � Classical extension in 2D of the Jin-Levermore scheme : modify the upwind fluxes (1D fluxes write in the normal direction) plugging the steady states in the fluxes. x r Cell Ω k x r + 1 x k l jk n jk x j Cell Ω j x r − 1 � l jk and n jk the normal and length associated with the edge ∂ Ω jk . Asymptotic limit of the scheme: E n k − E n | Ω j | ∂ t E j ( t ) − 1 j σ ∑ d ( x j , x k ) = 0. l jk k � || P 0 h − P h || → 0 only on strong geometrical conditions. � These AP schemes do not converge on 2D general meshes ∀ ε . 11 / 33 E. Franck FV nodal schemes 11/33

Example of unstructured meshes Random mesh Collela mesh 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Random triangular mesh Kershaw mesh 12 / 33 E. Franck FV nodal schemes 12/33

AP scheme for the P 1 model 13 / 33 E. Franck FV nodal schemes 13/33