Discrete entropy inequalities via an optimization process

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.