Error estimates of Runge–Kutta discontinuous galerkin methods for the Vlasov–Maxwell system∗
Revised: 29 April 2014
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 Chk + 1 / 2 + Cτ3 under a CFL condition τ/h ≤ γ. Here k is the polynomial degree used in phase space discretization, satisfying (with dx 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.
Mathematics Subject Classification: 65M15 / 65M60 / 65M06 / 35Q83 / 35L50
Key words: Vlasov–Maxwell system / Runge–Kutta discontinuous Galerkin methods / error estimates
© EDP Sciences, SMAI 2015