On the Relation Between the Upwind-Differencing Schemes of Godunov, Engquist–Osher and Roe

The upwind-differencing first-order schemes of Godunov, Engquist–Osher and Roe are discussed on the basis of the inviscid Burgers equations. The differences between the schemes are interpreted as differences between the approximate Riemann solutions on which their numerical flux-functions are based. Special attention is given to the proper formulation of these schemes when a source term is present. Second-order two-step schemes, based on the numerical flux-functions of the first-order schemes are also described. The schemes are compared in a numerical experiment and recommendations on their use are included.