SUMMARYThe numerical investigation of shock phenomena in gas or liquid media where enthalpy is the preferred thermodynamic variable poses special problems. When an expression for internal energy is available, the usual procedure is to employ a splitting scheme to remove source terms for the Euler equations, then upwind-biased shock capturing algorithms are built around the Riemann problem for the conservative system which remains. However, when the governing equations are formulated in terms of total enthalpy, treatment of a pressure-time derivative as a source term leads to a Riemann problem for a system where one equation is not a conservative law. The present research establishes that successful upwind-biased shock capturing schemes can be based upon the pseudoconservative system. The method is applied to numerical simulations of shock wave propagation in pure water.

In coastal oceanography there is interest in problems modeled by the shallow water equations, where variations in channel depth are accounted for by the presence of source terms. A numerical treatment for the solution of such problems is presented here, in terms of a hybrid approach, which combines a second-order TVD scheme for conservation law equations (assuming no source terms) with an eigenvector projection scheme that incorporates the effects of nonzero source terms (in regions where the bottom is not flat). For the case where an initially sharp wave profile is assumed, the progress of a wave as it traverses an estuary whose channel depth varies is calculated. Excellent numerical results are obtained.