Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws

Abstract

Entropy stability plays an important role in the dynamics of nonlinear systems of hyperbolic conservation laws and related convection-diffusion equations. Here we are concerned with the corresponding question of numerical entropy stability - we review a general framework for designing entropy stable approximations of such systems. The framework, developed in [28, 29] and in an ongoing series of works [30, 6, 7], is based on comparing numerical viscosities to certain entropy-conservative schemes. It yields precise characterizations of entropy stability which is enforced in rarefactions while keeping sharp resolution of shocks. We demonstrate this approach with a host of second- and higher-order accurate schemes, ranging from scalar examples to the systems of shallow-water, Euler and Navier-Stokes equations. We present a family of energy conservative schemes for the shallow-water equations with a well-balanced description of their steady-states. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in Euler equations, and we conclude with the computation of entropic measure-valued solutions based on the class of so-called TeCNO schemes - arbitrarily high-order accurate, non-oscillatory and entropy stable schemes for systems of conservation laws.