Some contributions to the modelling of discontinuous flows

Roe (1980) advanced the claim that 'almost any' good numerical technique for solving the linear scalar wave equation could be converted, by a fairly simple mechanism, into an equally good numerical technique for solving quasi-linear systems of conservation laws. It is pointed out that the particular attraction of that mechanism lay in its ability to generalize the class of asymmetric, or 'upwinded' finite-difference schemes, in a simple and natural way. In the present investigation, Roe's claim is reviewed and revised in the light of two and a half years of additional experience. A fairly general form of the flux difference splitting concept, applicable to one-dimensional conservation laws, is discussed. Attention is given to the 'monotone' second-order scheme, the designing of the B-functions, and problems regarding satisfying entropy.