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

TL;DR

This paper analyzes a finite volume scheme for 1D linear kinetic equations, proving it is asymptotic-preserving and hypocoercive, with uniform decay rates, supported by numerical simulations demonstrating accuracy and efficiency.

Contribution

It introduces an asymptotic-preserving finite volume scheme for kinetic equations and adapts hypocoercivity methods to the discrete setting for exponential convergence.

Findings

Scheme is asymptotic-preserving in the diffusive limit

Exponential decay to equilibrium with uniform rates

Numerical simulations confirm accuracy and efficiency

Abstract

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. Amer. Math. Soc., 367, 6 (2015)] to prove the exponential return to equilibrium of the approximate solution. We obtain decay rates that are bounded uniformly 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.