Sparse Grid Methods for Solving Vlasov-Poisson and Vlasov-Maxwell Systems

The Vlasov-Poisson and Vlasov-Maxwell models are fundamental to the kinetic treatment of plasma physics. Here we consider the fully-kinetic Vlasov equation which describes the time evolution of a distribution function in a six dimensional phase space (3x-3v), plus time, and is coupled either with the Poisson’s or Maxwell’s equations in three dimensions (3x). For continuum methods, the challenging issue is how to design an efficient and stable numerical scheme for handling such high dimensionality. In this work, we present a numerical scheme for approximating Vlasov-Poisson and Vlasov-Maxwell systems in a generalized dimensionality. Our approach is based on the multiwavelet basis and sparse grids, yielding significant reductions in the number of unknowns required for high dimensional approximation. We will present several standard benchmark tests, reporting on how their degree of freedoms and accuracy compare with those of the standard full grid approach.