Applied Math Seminar
Monday, September 27, 1999, 4:00pm, 120 Physics
J. Thomas Beale (Duke University)
Boundary Integral Methods for Water Waves
Abstract:
We discuss numerical methods for time-dependent water waves and the analysis needed to prove the convergence of properly designed versions to the actual solution. Methods in wide use are based on singular integrals arising from potential theory. The water surface is tracked by points which move with the fluid velocity. The velocity is determined from an integral equation on the surface. After reviewing the equations of motion, we will formulate the problem in boundary integrals and discuss some of the numerical issues. A central concern is the efficient computation of singular integrals such as single layer potentials on a surface. The approach here is to regularize the singularity, use a standard quadrature, and add a simple local correction. For the stability estimates, mapping properties of the discrete integral operators are used which are found by treating the sums as discrete versions of pseudodifferential operators.

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