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

TL;DR

This paper introduces a sparse grid discontinuous Galerkin method for high-dimensional transport equations, demonstrating its efficiency and accuracy in kinetic simulations with reduced computational costs.

Contribution

The paper develops a novel sparse grid DG scheme that is stable, convergent, and computationally efficient for high-dimensional transport problems.

Findings

Proven $L^2$ stability and convergence of the scheme.

Effective in simulating Vlasov and Boltzmann equations up to four dimensions.

Maintains accuracy and conservation properties in numerical tests.

Abstract

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