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

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.