An entropy fix for multi-dimensional upwind residual distribution schemes

Abstract

The residual distribution method allows a genuinely multi-dimensional generalization of the flux difference splitting schemes that are commonly used for dimension-by-dimension upwinding in finite volume discretization methods. Despite their many advantages, the residual distribution schemes also share an important deficiency with the Roe-type flux difference splitting schemes: they admit expansion shocks and thus do not respect the entropy condition, i.e. the second law of thermodynamics. In this paper the entropy condition violation of the residual distribution method is analyzed and an entropy fix is presented. By adding the entropy fix the deficiency is cured, as demonstrated by two examples of steady supersonic Euler flow.