The behavior of flux difference splitting schemes near slowly moving shock waves

Abstract

An investigation of the behavior of shock capturing schemes which compute the numerical flux from a solution of Riemann's problem is performed. The schemes of Godunov, Roe, and Osher are examined for a one-dimensional model problem consisting of a nearly stationary shock. Both scalar and systems of equations are examined. It is found that for slow shocks there is a significant error generated when solving systems of equations, while the scalar results are well behaved. This error consists of a long wavelength noise in the downstream running wave families that is not effectively damped by the dissipation of the scheme. The source of this error is shown, and the implications for the performance of these schemes are considered. This error may contribute to the slow convergence to steady state reported by many researchers.