Using the Dafermos Entropy Rate Criterion in Numerical Schemes

TL;DR

This paper develops a finite volume numerical scheme that is entropy stable according to Dafermos's criterion, combining entropy conservative and dissipative fluxes to achieve high accuracy and non-oscillatory shock behavior.

Contribution

It introduces a novel semidiscrete scheme that ensures entropy stability in the sense of Dafermos and classical entropy dissipation, with high-order accuracy and shock-capturing capabilities.

Findings

Achieves 2p order accuracy in smooth regions

Demonstrates non-oscillatory behavior around shocks

Ensures entropy stability according to Dafermos's criterion

Abstract

The following work concerns the construction of an entropy dissipative finite volume solver based on the convex combination of an entropy conservative and an entropy dissipative flux. We aim to construct a semidiscrete scheme that is entropy stable in the sense of the entropy criterion of Dafermos as well as in the classical sense entropy dissipative. The proposed semidiscrete scheme shows nice properties like $2p$ order accuracy in smooth regions as well as a non-oscillatory behavior around shocks.