Three-Dimensional Modeling of Potential Waves

Abstract

A simple and exact numerical scheme for long-term simulations of three-dimensional potential fully nonlinear periodic gravity waves is suggested. The scheme is based on the surface-following non-orthogonal curvilinear coordinate system. Velocity potential is represented as a sum of analytical and nonlinear components. The Poisson equation for the nonlinear component of velocity potential is solved iteratively. Fourier transform method, the second-order accuracy approximation of vertical derivatives on a stretched vertical grid, and the fourth-order Runge–Kutta time stepping are used. The scheme is validated by simulation of steep Stokes waves. A one-processor version of the model for PC allows us to simulate evolution of a wave field with thousands of degrees of freedom for hundreds of wave periods. The scheme is designed for investigation of nonlinear two-dimensional surface waves, generation of extreme waves, and direct calculations of nonlinear interactions.