We study a class of spatial discretizations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials. In particular, we focus on spectral methods and discontinuous Galerkin approximations. To obtain L 2 stability properties, we introduce a new L 2 weighted space, with a time dependent weight. For the Hermite spectral form of the Vlasov-Poisson system, we prove conservation of mass, momentum and total energy, as well as global stability for the weighted L 2 norm. These properties are then discussed for several spatial discretizations. Finally, numerical simulations are performed with the proposed DG/Hermite spectral method to highlight its stability and conservation features.

翻译：我们研究Vlasov-Poisson系统的一组空间分解,这些分解是用Hermite 多元海洋学作为双曲系统编写的,我们特别侧重于光谱方法和不连续的Galerkin近似值。为了获得L 2稳定性特性,我们采用了一个新的L 2加权空间,有时间依赖权重。对于Vlasov-Poisson系统的Hermite光谱形式,我们证明了对质量、动力和总能量的保护,以及加权L 2 规范的全球稳定性。这些特性随后讨论若干空间分解特性。最后,用拟议的DG/Hermite光谱方法进行了数字模拟,以突出其稳定性和保存特征。