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

TL;DR

This paper proves the convergence and stability of a Hermite spectral method combined with discontinuous Galerkin techniques for solving the Vlasov-Poisson system, providing rigorous error estimates.

Contribution

It introduces a weighted L2 space with a time-dependent weight to establish stability and convergence of the method for the Vlasov-Poisson system.

Findings

Global stability in weighted L2 norm

Error estimates between numerical and smooth solutions

Propagation of regularity for the numerical method

Abstract

We prove the convergence of discontinuous Galerkin approximations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials in velocity. To obtain stability properties, we introduce a suitable weighted L 2 space, with a time dependent weight, and first prove global stability for the weighted L 2 norm and propagation of regularity. Then we prove error estimates between the numerical solution and the smooth solution to the Vlasov-Poisson system.