A discontinuous Galerkin method for the Vlasov-Poisson system

TL;DR

This paper introduces a discontinuous Galerkin method for the Vlasov-Poisson system, demonstrating its accuracy, conservation properties, and effectiveness through various linear and nonlinear plasma simulations.

Contribution

The paper presents a novel discontinuous Galerkin approach that preserves key physical properties and accurately simulates plasma dynamics in the Vlasov-Poisson system.

Findings

Method conserves mass and positivity of distribution functions.

Accurately reproduces linear Landau damping and linear advection.

Effectively captures nonlinear phenomena like two-stream instability.

Abstract

A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.