The motion of deep water waves is a long-existing problem in fluid dynamics. Because it has many applications in academics and industries, numerous scientists have done much research to tackle this problem. The evolution of deep water wave is governed by Euler equations. Because of its nonlinearity, there are few available analytical expressions to fully describe the evolution of the deep water waves. Thus efficient numerical approximation becomes very critical to understand the properties of water wave motion.
An improved boundary integral technique is presented in this thesis to simulate the motion of deep water waves. One difficulty is the singularity in the Green’s function. Two methods to treat this singularity are discussed. One is blob regularization with third-order accuracy, and the other is based on polar coordinates with spectral accuracy. In the blob regularization, we replace the Green’s function by a regularized smooth Green’s function, which provides a good approximation to the original integral. For the other approach, integral identities are applied to reduce the strength of the singularity and then a polar coordinate transformation is applied to obtain a nonsingular integrand. The results from these two methods will be examined.
Another challenge is that the integrands are integrated over an infinite surface. For a doubly periodic water wave, we have to sum the images of Green’s function over the free-surface of the water. Ewald summation technique is used to expedite the calculation. Three-dimensional interpolation technique is suggested to reduce the time spent even further.
The numerical method is tested with several examples and then applied to the motion of a perturbed Stokes wave. The perturbation grows until the resolution fails.