Numerical methods for time-dependent water waves

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.