Entropy and the numerical integration of conservation laws

In this paper, we review recent results on the role of entropy in the numerical integration of conservation laws. It is well known that weak solutions of systems of conservation laws may not be unique. Physically relevant weak solutions possess a viscous profile and satisfy entropy inequalities. In the discrete case entropy inequalities are used as a tool to prove convergence to entropy dissipating weak solutions. We start with classical results, in which entropy stability is used to prove the convergence of numerical solutions. We continue with entropy stable schemes, which are designed in order to produce entropy dissipation and can be proven to be convergent. Next we consider entropy as an error/regularity indicator permitting a local control on the behavior of the scheme, and thus driving grid and scheme adaptivity.