Study of conservation and recurrence of Runge-Kutta discontinuous Galerkin schemes for Vlasov-Poisson systems

TL;DR

This paper investigates Runge-Kutta discontinuous Galerkin schemes for Vlasov-Poisson systems, highlighting their conservation, accuracy, positivity preservation, and recurrence properties through Fourier analysis and benchmark tests.

Contribution

It provides a rigorous Fourier analysis of recurrence phenomena and evaluates the effects of polynomial spaces and limiters on solution quality in RKDG methods.

Findings

RKDG schemes exhibit excellent conservation properties.

Fourier analysis reveals recurrence behavior in DG methods.

Benchmark tests demonstrate the effectiveness of the schemes in plasma simulations.

Abstract

In this paper we consider Runge-Kutta discontinuous Galerkin (RKDG) schemes for Vlasov-Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preserving limiter that guarantees positivity of the distribution functions. The RKDG solver for the Vlasov equation is the main focus, while the electric field is obtained through the classical representation by Green's function for the Poisson equation. A rigorous study of recurrence of the DG methods is presented by Fourier analysis, and the impact of different polynomial spaces and the positivity-preserving limiters on the quality of the solutions is ascertained. Several benchmark test problems, such as Landau damping, two-stream instability and the KEEN (Kinetic Electrostatic Electron Nonlinear) wave, are given.