Use of a rotated Riemann solver for the two-dimensional Euler equations

Abstract

A scheme for the two-dimensional Euler equations that uses flow parameters to determine the direction for upwind -differencing is described. This approach respects the multi-dimensional nature of the equations and reduces the grid-dependence of conventional schemes. Several angles are tested as the dominant upwinding direction, including the local flow and velocity-magnitude-gradient angles. Roe's approximate Riemann solver is used to calculate fluxes in the upwind direction, as well as for the flux components normal to the upwinding direction. The approach is first tested for two-dimensional scalar convection, where the scheme is shown to have accuracy comparable to a high-order MUSCL scheme. Solutions of the Euler equations are calculated for a variety of test cases. Substantial improvement in the resolution of shock and shear waves is realized. The approach is promising in that it uses flow solution features, rather than grid features, to determine the orientation for the solution method.