Approximate Riemann solutions of the shallow water equations

A finite difference scheme based on flux difference splitting is presented for the solution of the onedimensional shallow water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids non-physical, spurious oscillations. An extension to the one-dimensional equations with source terms, is included. The scheme for the one-dimensional equations is applied to the dam-break problem, and the approximate solution is compared to the exact solution of ideal fluid flow.