J. Thomas
Beale

Department of Mathematics

Duke University

**Abstract:**
We will discuss a class of numerical methods for water waves
of boundary integral type. Such methods have been in use for some
time in mathematics and ocean engineering. The evolution in time
of the free boundary and its velocity can be reduced to the
computation of quantities on the boundary, involving singular
integrals, provided the flow is irrotational. The approach is
Lagrangian; that is, the surface is represented by markers which
follow the flow. Numerical instablities have long been observed
and dealt with in these methods. The speaker, T. Y. Hou, and
J. S. Lowengrub have analyzed the errors in such methods and
designed versions for two-dimensional, periodic waves, which are
numerically stable and converge to the exact solution. The
analysis applies to the fully nonlinear regime. It is important
that the discrete equations have a structure analogous to
that of the continuous case. An example with an overturning
wave illustrates the ability to capture small-scale structure.