A TVD MacCormack scheme for transcritical flow Available to Purchase

A simple update to a well-known classical scheme for solving the non-linear shallow water equations is described which is computationally efficient and able to resolve accurately both subcritical and supercritical flows. The solver is the MacCormack scheme with a total variation diminishing term appended to the corrector step which eliminates spurious numerical oscillations which often arise when the convective terms are discretized using classical central difference schemes. The scheme is explicit, second-order in space and time and formulated in finite volume form for ease of implementation on a general boundary conforming grid. Bench mark solutions in one and two dimensions involving both steady and non-steady flows are shown to illustrate the high spatial accuracy and computational efficiency of the method compared to the MacCormack schemes and a modern Riemann-based upwind scheme.