Journal of Theoretical & Computational Science
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Mathematical Model of Complete Shallow Water Problem with Source Terms, Stability Analysis of Lax-Wendroff Scheme

Florence T Namio, Eric Ngondiep, Romaric Ntchantcho and Jean C Ntonga

The most effective simulations of complete physical problems consist of the evaluation of maximum water levels and discharges that may be attained at particular locations during the development of an exceptional meteorological event. There is also the prevision of the scenario subsequent to the almost instantaneous release of a great volume of liquid. The situation is that of the breaking of a man made dam. There is therefore a necessity to develop a model capable of reproducing solutions of the complete equations despite the irregularities of a non-prismatic bed. This requires the development of efficient and effective numerical schemes able to predict water levels and discharges in hydraulic systems. The use of mathematical models as a predictive tool in the simulation of free surface flows represents a good candidate for the application of many of the techniques developed in fluid dynamics. In this paper we develop a 1-D complete model of shallow water equations with source terms using both conservation of water mass and conservation of the momentum content of the water. We describe the Lax-Wendroff scheme for these nonlinear partial differential equations (PDEs) and we analyze the stability restriction of the method. This extends the nonstationary shallow water problems without source terms which are deeply studied in literature. Some numerical experiments are considered and critically discussed.