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

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