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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