Fully spectral scheme for the linear BGK equation on the whole space

TL;DR

This paper introduces a fully spectral numerical scheme for the linear BGK kinetic equation on the entire space, which preserves hypocoercivity and conservation laws, demonstrated through numerical experiments with polynomial potentials.

Contribution

The paper presents the first spectral method that maintains hypocoercive behavior and conservation laws for the linear BGK equation with polynomial confinement potential.

Findings

Scheme preserves hypocoercivity and conservation laws.

Numerical experiments validate the scheme on quadratic and double well potentials.

Applicable to various collision operators beyond BGK.

Abstract

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $\phi$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-\phi}$ in space. The potential $\phi$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.