On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws

TL;DR

This paper introduces a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems, proving its stability and convergence to entropy solutions without using streamline diffusion stabilization.

Contribution

The paper presents a novel entropy stable shock capturing DG scheme that guarantees convergence to entropy measure-valued solutions for general systems without SD stabilization.

Findings

Proves entropy stability and consistency of the scheme.

Establishes convergence to entropy measure-valued solutions.

Applicable to arbitrary order DG methods and general systems.

Abstract

In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [14], our approach is new in that we do not use streamline diffusion (SD) stabilization. It is proved that an artificial-viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.