Convergence of a Boundary Integral Method for 3-d Water Waves

We prove convergence of a modified point vortex method for timedependent water waves in a three-dimensional, inviscid, irrotational and incompressible fluid. Our stability analysis has two important ingredients. First we derive a leading order approximation of the singular velocity integral. This leading order approximation captures all the leading order contributions of the original velocity integral to linear stability. Moreover, the leading order approximation can be expressed in terms of the Riesz transform, and can be approximated with spectral accuracy. Using this leading order approximation, we construct a near field correction to stabilize the point vortex method approximation. With the near field correction, our modified point vortex method is linearly stable and preserves all the spectral properties of the continuous velocity integral to the leading order. Nonlinear stability and convergence with 3rd order accuracy are obtained using Strang’s technique by establishing an error expansion in the consistency error.