Boundary Perturbation Methods for Water Waves

The most successful equations for the modeling of ocean wave phenomena are the free– surface Euler equations. Their solutions accurately approximate a wide range of physical problems from open–ocean transport of pollutants, to the forces exerted upon oil platforms by rogue waves, to shoaling and breaking of waves in nearshore regions. These equations provide numerous challenges for theoreticians and practitioners alike as they couple the difficulties of a free boundary problem with the subtle balancing of nonlinearity and dispersion in the absence of dissipation. In this paper we give an overview of, what we term, “Boundary Perturbation” methods for the analysis and numerical simulation of this “water wave problem.” Due to our own research interests this review is focused upon the numerical simulation of traveling water waves, however, the extensive literature on the initial value problem and additional theoretical developments are also briefly discussed.