Entropy-stable finite difference and finite volume schemes for compressible flows

Abstract: In this talk, we present the main contributions made during the doctoral thesis work. We construct high-order finite difference schemes for systems of conservation laws on uniform Cartesian grids, which are provably entropy conservative/stable. In particular, we focus on the compressible flow equations, for which the proposed schemes are additionally able to approximate kinetic energy dynamics in a consistent manner. The entropy conservation ideas are extended to the rotating shallow water equations, to construct high-order energy preserving schemes. Finally, we propose a (formally) second-order entropy stable finite volume scheme for the Navier-Stokes equation on unstructured triangular grids, which are useful for problems involving complex domains. The robustness of the proposed schemes is demonstrated via several standard test cases.