Discrete hypocoercivity for a nonlinear kinetic reaction model

Marianne Bessemoulin-Chatard (1) , Tino Laidin (2, 3) , Thomas Rey (4)

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

In this article, we propose a finite volume discretization of a one dimensional nonlinear reaction kinetic model proposed in [Neumann, Schmeiser, Kint. Rel. Mod. 2016], which describes a 2-species recombination-generation process. Specifically, we establish the long-time convergence of approximate solutions towards equilibrium, at exponential rate. The study is based on an adaptation for a discretization of the linearized problem of the $L^2$ hypocoercivity method introduced in [Dolbeault, Mouhot, Schmeiser, 2015]. From this, we can deduce a local result for the discrete nonlinear problem. As in the continuous framework, this result requires the establishment of a maximum principle, which necessitates the use of monotone numerical fluxes.

Mots clés: kinetic equations; hypocoercivity; maximum principle; finite volume methods; large time behavior

Domaines: Analyse numérique [math.NA]

Marianne Bessemoulin-Chatard, Tino Laidin, Thomas Rey. Discrete hypocoercivity for a nonlinear kinetic reaction model. 2024. ⟨hal-04494454⟩