Contribution on Numerical Solutions of the Shallow-Water Equation

Abstract

The one-dimensional shallow-water equation is used to simulate the flood wave in open channel. Two examples have been considered, the former includes routing a discharge hydrograph down a rectangular channel. The second includes routing a sudden shutoff of all flow at the downstream end of the channel. Some numerical methods are developed: the method of characteristics, using non-linear interpolation, the Mac-Cormack and the optimal finite difference scheme. Surge wave were considered in a sloped and frictional, rectangular channel. Validity of numerical models is obvious compared with the experimental data. An additional comparison reveals good match between the obtained computed results and those from the quoted literature. It has been shown that the Mac-Cormack scheme is robust for modelling both subcritical and supercritical flows in open channels. Moreover numerical oscillations, produced by this method, are avoided when using the optimal scheme. This is of special importance since these conditions present serious problems for numerical algorithms using the method of characteristics. Good accuracy is achieved in these two classes of the finite difference method, including the automatic capturing of shock waves. However, the finite difference schemes are deemed unsuitable for simulating high roughness channel due to the requirement of very small time steps.