Stable, high-order computation of traveling water waves in three dimensions

The Euler equations of free-surface ocean dynamics constitute a model of central importance in fluid mechanics due to the wide range of physical phenomena they are intended to represent, from shoaling and breaking of waves in nearshore regions to energy and momentum transport in the open ocean. From a mathematical perspective, these equations present rather unique challenges for analysis and simulation as they couple the subtleties of nonlinear wave equations (balancing nonlinearity with dispersion in the absence of dissipation) to the difficulties of free-boundary problems. In this paper a new, stable high-order boundary perturbation algorithm for the numerical simulation of traveling water waves is described. Its performance is compared to that of classical surface deformation algorithms and it is shown that the new scheme displays significantly enhanced conditioning properties and a lower computational cost, which enable very accurate predictions of physical observables such as velocity, energy, height/steepness, and shape.