An adaptive multiresolution discontinuous Galerkin method with artificial viscosity for scalar hyperbolic conservation laws in multidimensions

TL;DR

This paper introduces an adaptive multiresolution discontinuous Galerkin method with artificial viscosity for solving scalar hyperbolic conservation laws in multiple dimensions, improving efficiency and stability in capturing shocks.

Contribution

It develops a novel adaptive multiresolution DG scheme using interpolatory multiwavelets and artificial viscosity, enhancing computational efficiency and shock capturing capabilities.

Findings

Achieves similar complexity as sparse grid DG for smooth solutions

Demonstrates high accuracy and robustness in numerical tests

Effectively captures shocks with added artificial viscosity

Abstract

In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for scalar hyperbolic conservation laws in multidimensions. Compared with previous work for linear hyperbolic equations \cite{guo2016transport, guo2017adaptive}, a class of interpolatory multiwavelets are applied to efficiently compute the nonlinear integrals over elements and edges in DG schemes. The resulting algorithm, therefore can achieve similar computational complexity as the sparse grid DG method for smooth solutions. Theoretical and numerical studies are performed taking into consideration of accuracy and stability with regard to the choice of the interpolatory multiwavelets. Artificial viscosity is added to capture the shock and only acts on the leaf elements taking advantages of the multiresolution representation. Adaptivity is realized by auto error thresholding based on hierarchical surplus. Accuracy and robustness are demonstrated by several numerical tests.