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.

翻译：有限体积方法是求解时间相关偏微分方程（尤其是双曲守恒律）的常用工具。过去40年间，增强其稳健性的主流途径是强制满足全局或局部熵不等式。本研究聚焦于Dafermos近50年前提出的一个不同熵准则：该准则要求选取的弱解能以最快速度耗散选定熵。我们证明，将该熵率准则与最优恢复理论相结合时，可应用于数值计算。迄今为止，该准则在有限体积格式中应用有限，据作者所知，本研究首次将此准则应用于基于均值重构的精度型有限体积格式。由此催生出一类基于重构的新型格式，为经典ENO和WENO格式提供了替代方案。