Modified method of characteristics for the shallow water equations

Flow in open channels is frequently modelled using the shallow water equations (SWEs) with an up-winded scheme often used for the nonlinear terms in the numerical scheme (Delis et al., 2000; Erduran et al., 2002). This paper presents a mathematical model based on the SWEs to compute one dimensional (1-D) open channel flow. Two techniques have been used for the simulation of the flood wave along streams which are initially dry. The first one uses up-winding applied to the convective acceleration term in the SWEs to overcome the problem of numerical instabilities. This is applied to the integration of the shallow water equations within the domain, so the scheme does not require any special treatment, such as artificial viscosity or front tracking technique, to capture steep gradients in the solution. As in all initial value problems, the main difficulty is the boundaries, the conventional method of characteristics (MOC) can be applied in a straight forward way for a lot of cases, but when dealing with a very shallow initial depths followed by a flood wave, it is not possible to overcome the problem of reflections. So a modified method of characteristics (MMOC) is the second technique that has been developed by the authors to obtain a fully transparent downstream boundary and is the main subject of this paper. The mathematical model which integrates the SWEs using a staggered finite difference scheme within the domain and the MMOC near the boundary has been tested not only by comparing its results with some analytical solutions for both steady and unsteady flow but also by comparing the results obtained with the results of other models such as Abiola et al. (1988).