On the Convergence of Space-Time Discontinuous Galerkin Schemes for Scalar Conservation Laws

TL;DR

This paper proves the convergence of space-time discontinuous Galerkin schemes for scalar conservation laws, showing they reach the unique entropy solution without the need for streamline-diffusion stabilization, for all polynomial orders.

Contribution

It demonstrates convergence of DG schemes to entropy solutions without streamline-diffusion stabilization, aligning with practical implementations and extending previous theoretical results.

Findings

Convergence holds for all polynomial approximation orders.

No streamline-diffusion stabilization is necessary.

Applicable to schemes with strictly monotone flux functions.

Abstract

We prove convergence of a class of space-time discontinuous Galerkin schemes for scalar hyperbolic conservation laws. Convergence to the unique entropy solution is shown for all orders of polynomial approximation, provided strictly monotone flux functions and a suitable shock-capturing operator are used. The main improvement, compared to previously published results of similar scope, is that no streamline-diffusion stabilization is used. This is the way discontinuous Galerkin schemes were originally proposed, and are most often used in practice.