AbstractsMathematics

The motion of free surface flow has been widely studied in the scientific community. It has applications in both civil and military engineering. Through years of research, we can now formulate mathematical models to express the phenomena such as ocean waves, rain drops, waterfalls etc. The purpose of this thesis is to study the velocity potential of shallow water wave. The motions of shallow water, unlike deep water, are influenced by the bottom topology. We apply boundary integral technique and solve a system of two integral equations, one expresses the surface energy potential, another expresses the influence of the bottom topology. Boundary integrals can produce very accrue results for shallow water equations but they are too complex to solve analytically. Therefore, in this thesis, we developed a computer program to give us numerical results. We use three different numerical scheme to compute the solutions of this integral equation system. They are Gaussian elimination, standard Jacobi iteration and Jacobi iteration with preconditioner. Remarkably, they all converge to the same results, which means that we are getting the right solutions. The computation cost is very high for Gaussian Elimination and Jacobi iteration. In order to lower the cost, a preconditioner is applied to Jacobi iterative scheme. For shallow depth, the preconditioner effectively lowers the total computation cost.