An Adaptive Multiresoluton Discontinuous Galerkin Method for Time-Dependent Transport Equations in Multi-dimensions

TL;DR

This paper introduces an adaptive multiresolution discontinuous Galerkin method for efficiently solving multi-dimensional, time-dependent transport equations, capable of capturing local structures and reducing computational costs.

Contribution

It develops a novel adaptive multiresolution DG scheme using multiwavelets and error thresholding, enhancing efficiency and accuracy in multi-dimensional kinetic simulations.

Findings

Performs comparably to sparse grid DG for smooth solutions

Automatically captures local structures in non-smooth solutions

Reduces computational cost in multi-dimensions

Abstract

In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for time-dependent transport equations in multi-dimensions. The method is constructed using multiwavlelets on tensorized nested grids. Adaptivity is realized by error thresholding based on the hierarchical surplus, and the Runge-Kutta DG (RKDG) scheme is employed as the reference time evolution algorithm. We show that the scheme performs similarly to a sparse grid DG method when the solution is smooth, reducing computational cost in multi-dimensions. When the solution is no longer smooth, the adaptive algorithm can automatically capture fine local structures. The method is therefore very suitable for deterministic kinetic simulations. Numerical results including several benchmark tests, the Vlasov-Poisson (VP) and oscillatory VP systems are provided.