A Second-Order Unsplit Godunov Scheme for Two- and Three-Dimensional Euler Equations

Abstract

A second-order finite difference scheme is presented in this paper for two- and three-dimensional Euler equations. The scheme is based on nondirectionally split and single-step Eulerian formulations of Godunov approach. A new approach is proposed for constructing effective left and right states of Riemann problems arising from interfaces of one-, two-, and three-dimensional numerical grids. The Riemann problems are solved through an approximate solver in order to calculate a set of time-averaged fluxes needed in the scheme. The scheme is tested through numerical examples involving strong shocks. It is shown that the scheme offers the principle advantages of a second order Godunov scheme: robust operation in the presence of strong waves and thin shock fronts.