On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system

TL;DR

This paper analyzes the stability and conservation properties of Hermite spectral and discontinuous Galerkin methods for the Vlasov-Poisson system, introducing a new weighted L2 space to ensure stability.

Contribution

It introduces a novel weighted L2 space with a time-dependent weight to establish stability for Hermite spectral methods applied to the Vlasov-Poisson system.

Findings

Proves conservation of mass, momentum, and energy for the spectral form.

Establishes global stability in the weighted L2 norm.

Numerical simulations confirm the stability and conservation properties.

Abstract

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.