Entropy stability of Roe-type upwind finite volume methods on unstructured grids

One reason which makes Roe’s Riemann solver attractive is its low computational cost. But the main drawback with Roe’s approximate Riemann solver is that non-physical expansion shocks can occur in the sonic points, it has been early remarked that for this particular situation. The Roe flux does not satisfy the entropy condition. In this paper an elegant response has been proposed by combining Harten and Tadmor entropy correction, with tuning parameters that play in fact the role of an artificial viscosity. Convergence and consistency with the entropy condition are proved. Numerical results on a two–dimensional hypersonic flow around a blunt body and double ellipsöıd confirm the theoretical results.