Error Estimates of Runge-Kutta Discontinuous Galerkin Methods for the Vlasov-Maxwell System

TL;DR

This paper establishes error bounds for Runge-Kutta discontinuous Galerkin methods applied to the Vlasov-Maxwell system, providing theoretical guarantees for the accuracy of plasma simulations under certain regularity assumptions.

Contribution

It provides the first rigorous error analysis for RKDG methods solving the Vlasov-Maxwell system, including error bounds and conditions for convergence.

Findings

Error bounds of order $h^{k+1/2} + au^3$ for particle density and fields.

Error estimates depend on polynomial degree, mesh size, and time step.

Analysis applicable to various fluxes and relativistic equations.

Abstract

In this paper, error analysis is established for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve the Vlasov-Maxwell system. This nonlinear hyperbolic system describes the time evolution of collisionless plasma particles of a single species under the self-consistent electromagnetic field, and it models many phenomena in both laboratory and astrophysical plasmas. The methods involve a third order TVD Runge-Kutta discretization in time and upwind discontinuous Galerkin discretizations of arbitrary order in phase domain. With the assumption that the exact solution has sufficient regularity, the $L^2$ errors of the particle number density function as well as electric and magnetic fields at any given time $T$ are bounded by $C h^{k+\frac{1}{2}}+C\tau^3$ under a CFL condition $\tau /h \leq \gamma$. Here $k$ is the polynomial degree used in phase space discretization, satisfying $k \geq \left \lceil \frac{d_x + 1}{2} \right \rceil$ (the smallest integer greater than or equal to $\frac{d_x+1}{2}$, with $d_x$ being the dimension of spatial domain), $\tau$ is the time step, and $h$ is the maximum mesh size in phase space. Both $C$ and $\gamma$ are positive constants independent of $h$ and $\tau$, and they may depend on the polynomial degree $k$, time $T$, the size of the phase domain, certain mesh parameters, and some Sobolev norms of the exact solution. The analysis can be extended to RKDG methods with other numerical fluxes and to RKDG methods solving relativistic Vlasov-Maxwell equations.