Discontinuous Galerkin Methods for the Vlasov-Maxwell Equations

TL;DR

This paper introduces discontinuous Galerkin methods for the Vlasov-Maxwell system that achieve high accuracy and conservation properties, with proven charge and energy conservation and verified effectiveness through numerical tests.

Contribution

It develops a systematic discontinuous Galerkin scheme for the Vlasov-Maxwell equations that ensures conservation and accuracy, with error estimates and validation.

Findings

Charge is conserved up to boundary effects.

Total energy can be preserved with proper flux choices.

The method's accuracy and conservation are verified on the Weibel instability.

Abstract

Discontinuous Galerkin methods are developed for solving the Vlasov-Maxwell system, methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Maxwell system. The proposed scheme employs discontinuous Galerkin discretizations for both the Vlasov and the Maxwell equations, resulting in a consistent description of the distribution function and electromagnetic fields. It is proven, up to some boundary effects, that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the Maxwell equations and the underlying approximation spaces. Error estimates are established for several flux choices. The scheme is tested on the streaming Weibel instability: the order of accuracy and conservation properties of the proposed method are verified.