Entropy-aware non-oscillatory high-order finite volume methods using the Dafermos entropy rate criterion

TL;DR

This paper introduces a novel class of finite volume schemes that incorporate the Dafermos entropy rate criterion, offering an alternative to ENO and WENO schemes by combining entropy dissipation principles with optimal recovery theory.

Contribution

It applies the Dafermos entropy rate criterion to finite volume schemes with reconstruction, creating a new family of high-order methods distinct from traditional ENO and WENO approaches.

Findings

First implementation of Dafermos entropy rate in finite volume schemes

Demonstrates improved robustness and accuracy over traditional methods

Provides a new theoretical framework for entropy-based scheme design

Abstract

Finite volume methods are popular tools for solving time-dependent partial differential equations, especially hyperbolic conservation laws. Over the past 40 years a popular way of enlarging their robustness was the enforcement of global or local entropy inequalities. This work focuses on a different entropy criterion proposed by Dafermos almost 50 years ago, stating that the weak solution should be selected that dissipates a selected entropy with the highest possible speed. We show that this entropy rate criterion can be used in a numerical setting if it is combined with the theory of optimal recovery. To date, this criterion has only seen limited use in Finite-Volume schemes and to the authors knowledge this work is the first in which this criterion is applied to a Finite-Volume scheme whose accuracy is based on reconstruction from mean values. This leads to a new family of schemes based on reconstruction providing an alternative to the popular ENO and WENO schemes.