A non-parameterized entropy correction for Roe's approximate Riemann solver

Abstract

The main drawback with Roe's approximate Riemann solver is that non-physical expansion shocks can occur in the vicinity of sonic points. Previous work aimed at enforcing the entropy condition is based on the representation of sonic rarefaction waves. We propose a new non-parameterized approach which is based on a nonlinear Hermite interpolation of an approximate flux function and the exact resolution of non convex scalar Riemann problems. Convergence and consistency with the entropy condition are proved for scalar convex conservation laws with arbitrarily large initial data. When considering strictly hyperbolic systems of conservation laws, consistency of the resulting scheme with the entropy condition is also proved for initial data sufficiently close to a constant. Numerical results on a one-dimensional shock-tube and a two-dimensional supersonic forward facing step confirm our theoretical results.