Entropy stable discontinuous Galerkin methods for nonlinear conservation laws on networks and multi-dimensional domains

TL;DR

This paper introduces a high-order entropy stable discontinuous Galerkin method for nonlinear conservation laws applicable to multi-dimensional domains and networks, ensuring stability at interfaces and junctions with verified numerical experiments.

Contribution

The paper develops a novel entropy stable DG method for complex geometries and networks, maintaining stability across interfaces and junctions, with demonstrated accuracy and stability.

Findings

Numerical experiments confirm the stability of the proposed schemes.

Comparisons show the accuracy of different junction treatments.

The methods effectively handle multi-dimensional and network domains.

Abstract

We present a high-order entropy stable discontinuous Galerkin (ESDG) method for nonlinear conservation laws on both multi-dimensional domains and on networks constructed from one-dimensional domains. These methods utilize treatments of multi-dimensional interfaces and network junctions which retain entropy stability when coupling together entropy stable discretizations. Numerical experiments verify the stability of the proposed schemes, and comparisons with fully 2D implementations demonstrate the accuracy of each type of junction treatment.