Energy-conserving Discontinuous Galerkin Methods for the Vlasov-Maxwell System

Abstract: In this paper, we generalize the idea in our previous work for the Vlasov-Amp`{e}re (VA) system \cite{cheng_va} and develop energy-conserving discontinuous Galerkin (DG) methods for the Vlasov-Maxwell (VM) system. The VM system is a fundamental model in the simulation of collisionless magnetized plasmas. Compared to \cite{cheng_va}, additional care needs to be taken for both the temporal and spatial discretizations to achieve similar type of conservation when the magnetic field is no longer negligible. Our proposed schemes conserve the total particle number and the total energy at the same time, and therefore can obtain accurate, yet physically relevant solutions. The main components of our methods include second order and above, explicit or implicit energy-conserving temporal discretizations, and DG methods for Vlasov and Maxwell's equations with carefully chosen numerical fluxes. Benchmark numerical tests such as the streaming Weibel instability are provided to validate the accuracy and conservation of the schemes.