Stabilizing discontinuous Galerkin methods using Dafermos' entropy rate criterion

TL;DR

This paper introduces a new stabilization technique for discontinuous Galerkin methods using Dafermos' entropy rate criterion, improving stability and entropy compliance through optimization-based corrections.

Contribution

It presents a novel stabilization approach based on solving optimization problems that enforce Dafermos' entropy criterion, offering an alternative to traditional shock capturing methods.

Findings

Enhanced stability over basic DG methods.

Retention of the scheme's order in numerical tests.

Ability to enforce classical entropy inequalities.

Abstract

A novel approach for the stabilization of the discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. The approach is centered around the efficient solution of linear or nonlinear optimization problems in every timestep as a correction to the basic discontinuous Galerkin scheme. The thereby enforced Dafermos criterion results in improved stability compared to the basic method while retaining the order of the method in numerical experiments. Further modification of the optimization problem allows also to enforce classical entropy inequalities for the scheme. The proposed stabilization is therefore an alternative to flux-differencing, finite-volume subcells, artificial viscosity, modal filtering, and other shock capturing procedures.