Iterative Solution of Boundary Integral Equations for Shallow Water Waves

The motion of shallow water, unlike deep water, is influenced by the bottom topology. In order to obtain the velocity potential at any given point of the surface at any given time, we apply boundary integral techniques and solve a system of two integral equations, one expresses the velocity potential at the free surface and on the bottom, the other expresses the streamfunction at the free surface and on the bottom. Boundary integrals can produce very accurate results for shallow water equations but they are too complex to solve analytically and must be solved numerically. In this paper, we use three different numerical methods to compute the solutions to these integral equations: by Gaussian elimination which has high compuational cost; by iteration to construct the Neumann series which converges; and by a preconditioned version of the iteration which effectively lowers the total computation cost