On a discrete framework of hypocoercivity for kinetic equations

TL;DR

This paper introduces a fully discrete finite volume scheme for the Vlasov-Fokker-Planck equation, leveraging Hermite polynomials to preserve key properties and establish convergence to equilibrium with asymptotic preservation in the diffusive limit.

Contribution

It develops a novel discrete scheme that maintains stationary solutions and entropy, and provides quantitative hypocoercivity-based convergence estimates for the kinetic equation.

Findings

Scheme preserves stationary solutions and entropy.

Provides explicit convergence rates to equilibrium.

Asymptotic preserving in the diffusive limit.

Abstract

We propose and study a fully discrete finite volume scheme for the Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted L 2 relative entropy. Then, we adapt the arguments developed in [12] based the hypocoercivity method to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play.