Error estimates of Runge–Kutta discontinuous galerkin methods for the Vlasov–Maxwell system^∗

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, 12180-3590 New York, USA.
yangh8@rpi.edu; lif@rpi.edu

Received: 3 December 2013
Revised: 29 April 2014

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 solutions have sufficient regularity, the L² errors of the particle number density function as well as electric and magnetic fields at any given time T are bounded by Ch^{k + 1 / 2} + Cτ³ under a CFL condition τ/h ≤ γ. Here k is the polynomial degree used in phase space discretization, satisfying $\mathit{k}\mathit{>}\frac{{\mathit{d}}_{\mathit{x}}\mathrm{+}\mathrm{1}}{\mathrm{2}}$ (with d_x being the dimension of spatial domain), τ is the time step, and h is the maximum mesh size in phase space. Both C and γ are positive constants independent of h and τ, 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.