In Section V, the free surface is perturbed by a periodic wave of small amplitude ɛ and the corrections to the eigenvalues and eigenvectors are calculated to order ɛ 2. However, the results can be used to design a preconditioner. In section IV, the eigenvalues and eigenvectors are determined exactly, and the spectral radius of the iteration matrix is found to be exp( -H), which means the rate of convergence is very slow for very shallow water. In order to study the rate of convergence of the iterations to construct the Neumann series, the eigenvalues and eigenvectors of the discrete matrix are calculated. In Section III, a numerical approximation for the integrals is introduced and the integral equations are replaced by discrete matrix equations. As a result of the formulation, two coupled Fredholm integral equations of the second kind must be solved to determine the dipole distributions. For two-dimensional flow, assumed in this paper, the integrals are performed along the water surface, a curve in two-dimensional space and along the boundary, also just a curve.
In Section II, the velocity potential and the streamfunction are expressed through dipole distributions along the water surface and along the bottom boundary. We follow in presenting integral equations for shallow water with high Reynolds numbers. Among the many applications, boundary integral methods calculate the velocity of incompressible fluid well for very small Reynolds numbers, which is discussed in, and very large Reynolds numbers, which is discussed in. The advantage of boundary integral techniques is that they only involve information on the free surface and boundary location, thus essentially reducing the dimensions of the problem by one.
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