Traveling Water Waves: Spectral Continuation Methods with Parallel Implementation

We present a numerical continuation method for traveling wave solutions to the full water wave problem using a spectral collocation discretization. The water wave problem is reformulated in terms of surface variables giving rise to the Zakharov–Craig–Sulem formulation, and traveling waves are studied by introducing a phase velocity vector as a parameter. We follow non-trivial solution branches bifurcating from the trivial solution branch via numerical continuation methods. Techniques such as projections and filtering allow the computation to proceed for greater distances up the branch, and parallelism allows the computation of larger problems. We conclude with results including the formation of hexagonal patterns for the three dimensional problem.