An efficient model for three-dimensional surface wave simulations - Part 1: Free space problems

An efficient numerical scheme for simulations of fully nonlinear non-breaking surface water waves in 3D is presented. The water depth is either shallow, finite or infinite. The method is based on a fast, rapidly converging, iterative algorithm to compute the Dirichlet to Neumann operator. This is evaluated by expanding the operator as a sum of global convolution terms and local integrals with kernels that decay quickly in space. The global terms are computed very quickly via FFT. The local terms are evaluated by numerical integration. Analytical integration of the linear part of the prognostic equations in Fourier space is obtained to machine precision. The remaining nonlinear components are integrated forward in time using an RK-scheme combined with a special step size control technique. This yields a very stable and accurate time marching procedure. Zeros-padding in the spectral space represents the anti-aliasing strategy. The method requires no smoothing. Illustration through examples show that the total energy is well conserved during the numerical simulations. The scheme is stable and accurate, even for very long time simulations of very steep wave events. The scheme is easily parallelizable. It propagates for example a Stokes wave of slope 0.2985 with a phase shift error of about 0.3 degrees after 1000 periods of propagation. (c) 2004 Elsevier Inc. All rights reserved.