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.

翻译：本文针对具有质量、动量和能量守恒的线性非均匀动力学方程，设计了一种在空间和速度维度上均采用谱方法求解的数值格式。我们聚焦于含约束势$\phi$的线性BGK方程，尽管该方法可推广至不同类型的碰撞算子。该方法的构建基于速度方向上的埃尔米特多项式投影，以及空间方向上相对于权函数$e^{-\phi}$的正交多项式展开。假设势函数$\phi$为多项式形式。据作者所知，这是首个在满足守恒律的同时还能保持亚椭圆耗散特性的数值格式。我们通过数值实验，在二次势阱和双势阱两种情形下验证了该格式的各项特性。