Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations

Marianne Chatard Bessemoulin-Chatard (1) , Maxime Herda (2) , Thomas Rey (2, 3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 LPP - Laboratoire Paul Painlevé - UMR 8524

In this article, we are interested in the asymptotic analysis of a finite volume scheme for one dimensional linear kinetic equations, with either Fokker-Planck or linearized BGK collision operator. Thanks to appropriate uniform estimates, we establish that the proposed scheme is Asymptotic-Preserving in the diffusive limit. Moreover, we adapt to the discrete framework the hypocoercivity method proposed by [J. Dolbeault, C. Mouhot and C. Schmeiser, Trans. Amed. Math. Soc., 367, 6 (2015)] to prove the exponential return to equilibrium of the approximate solution. We obtain decay estimates that are uniform in the diffusive limit. Finally, we present an efficient implementation of the proposed numerical schemes, and perform numerous numerical simulations assessing their accuracy and efficiency in capturing the correct asymptotic behaviors of the models.

Mots clés: Kinetic equations; finite volume methods; hypocoercivity; diffusion limit; asymptotic-preserving schemes

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques [math]; Analyse numérique [math.NA]

Marianne Chatard Bessemoulin-Chatard, Maxime Herda, Thomas Rey. Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations. 2018. ⟨hal-01957832v1⟩