Sparse Grid Discontinuous Galerkin Methods for the Vlasov-Maxwell System

Abstract

In this paper, we develop sparse grid discontinuous Galerkin (DG) schemes for the Vlasov-Maxwell (VM) equations. The VM system is a fundamental kinetic model in plasma physics, and its numerical computations are quite demanding, due to its intrinsic high-dimensionality and the need to retain many properties of the physical solutions. To break the curse of dimensionality, we consider the sparse grid DG methods that were recently developed in \cite{guo2016sparse,guo2017adaptive} for transport equations. Such methods are based on multiwavelets on tensorized nested grids and can significantly reduce the numbers of degrees of freedom. We formulate two versions of the schemes: sparse grid DG and adaptive sparse grid DG methods for the VM system. Their key properties and implementation details are discussed. Accuracy and robustness are demonstrated by numerical tests, with emphasis on comparison of the performance of the two methods, as well as with their full grid counterparts.