An efficient numerical scheme for the shallow water equations

Abstract

A finite difference scheme based on flux difference splitting is presented for the solution of the one-dimensional 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. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency. This is in contrast to the usual ”square root“ averages found in this type of Riemann solver where the computational expense can be prohibitive. 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.

Keywords:

• Shallow water equations
• flux difference splitting
• dam-break problem

C.R. CATEGORIES::

• G.1.8