A sparse grid discontinuous Galerkin method for high-dimensional transport equations and its application to kinetic simulations

摘要

In this paper, we develop a sparse grid discontinuous Galerkin (DG) schemefor transport equations and applied it to kinetic simulations. The method usesthe weak formulations of traditional Runge-Kutta DG (RKDG) schemes forhyperbolic problems and is proven to be $L^2$ stable and convergent. A majoradvantage of the scheme lies in its low computational and storage cost due tothe employed sparse finite element approximation space. This attractive featureis explored in simulating Vlasov and Boltzmann transport equations. Goodperformance in accuracy and conservation is verified by numerical tests in upto four dimensions.