Discrete entropy inequalities via an optimization process

¹ Sorbonne-Université and INRIA Paris, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), 75005 Paris, France
² Université Sorbonne Paris Nord and INRIA Paris, CNRS, Laboratoire Analyse, Géométrie et Applications (LAGA), 99 av. J.-B. Clément, 93430 Villetaneuse, France
³ Université de Picardie Jules Verne, CNRS, LAMFA, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France
⁴ INRIA Paris, ANGE Project-Team, 75589 Paris Cedex 12, France and Laboratoire Jacques-Louis Lions, Sorbonne Université, CNRS, 75005 Paris, France

^* Corresponding author: nina.aguillon@sorbonne-universite.fr

Received: 19 May 2023
Accepted: 30 November 2023

Abstract

The solutions of hyperbolic systems may contain discontinuities. These weak solutions verify not only the original PDEs, but also an entropy inequality that acts as a selection criterion determining whether a discontinuity is physical or not. Obtaining a discrete version of these entropy inequalities when approximating the solutions numerically is crucial to avoid convergence to unphysical solutions or even unstability. However such a task is difficult in general, if not impossible for schemes of order 2 or more. In this paper, we introduce an optimization framework that enables us to quantify a posteriori the decrease or increase of entropy of a given scheme, locally in space and time. We use it to obtain maps of numerical diffusion and to prove that some schemes do not have a discrete entropy inequality. A special attention is devoted to the widely used second order MUSCL scheme for which almost no theoretical results are known.