Discontinuous Galerkin Methods for Relativistic Vlasov–Maxwell System

The relativistic Vlasov–Maxwell (RVM) system is a kinetic model that describes the dynamics of plasma when the charged particles move in the relativistic regime and their collisions are not important. In this paper, we formulate and investigate discontinuous Galerkin (DG) methods to solve the RVM system. When standard piecewise polynomial functions are used to define trial and test spaces, the methods conserve mass as expected. However the energy conservation does not hold due to the specific form of the total energy of the system. In order to obtain provable mass and energy conservation, we take advantage of the flexibility of DG discretizations and enrich the discrete spaces with some non-polynomial function. For the semi-discrete DG methods with standard and enriched spaces, stability and error estimates are established together with their properties in conservation. In actual implementation with the enriched space, special care is needed to reduce the loss of significance for better numerical stability. Numerical experiments, including streaming Weibel instability and wakefield acceleration, are presented to demonstrate the performance of the methods. Positivity-preserving limiter is also used in simulating wakefield acceleration to obtain physically more relevant solutions.