A posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws

TL;DR

This paper develops reliable a posteriori error estimates for discontinuous Galerkin schemes applied to nonlinear hyperbolic conservation laws, using reconstruction techniques and relative entropy stability, validated through numerical tests.

Contribution

It introduces a general methodology for a posteriori error control of DG schemes for conservation laws using reconstructions and entropy frameworks.

Findings

Estimator effectively controls errors in DG schemes

Method is applicable to standard flux choices

Numerical tests confirm robustness of the estimator

Abstract

In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.