The Riemann Solver of Roe

Abstract

Perhaps, the most well—known of all approximate Riemann solvers today, is the one due to Roe, which was first presented in the open literature in 1981 [281]. Since then, the method has not only been refined, but it has also been applied to a very large variety of physical problems. Refinements to the Roe approach were introduced by Roe and Pike [290], whereby the computation of the necessary items of information does not explicitly require the Roe averaged Jacobian matrix. This second methodology appears to be simpler and is thus useful in solving the Riemann problem for new, complicated sets of hyperbolic conservations laws, or for conventional systems but for complex media. Glaister exploited the Roe—Pike approach to extend Roe’s method to the time—dependent Euler equations with a general equation of state [137], [138]. The large body of experience accumulated by many workers over a considerable period of time has led to various improvements of the scheme. As originally presented the Roe scheme computes rarefaction shocks, thus violating the entropy condition. Harten and Hyman [163], Roe and Pike [290], Roe [288], Dubois and Mehlman [113] and others, have produced appropriate modifications to the scheme. Einfeldt et. al. [118] produced corrections to the basic Roe scheme to avoid the so—called vacuum problem near low—density flows; they also showed that in fact this anomaly afflicts all linearised Riemann solvers.