A hybrid spectral/DG method for solving the phase-averaged ocean wave equation: Algorithm and validation
Journal of Computational Physics
We develop a new high-order hybrid discretization of the phased-averaged (action balance) equation to simulate ocean waves. We employ discontinuous Galerkin (DG) discretization on an unstructured grid in geophysical space and Fourier-collocation along the directional and frequency coordinates. The original action balance equation is modified to facilitate absorbing boundary conditions along the frequency direction; this modification enforces periodicity at the frequency boundaries so that the
... st convergence of Fourier-collocation holds. In addition, a mapping along the directional coordinate is introduced to cluster the collocation points around steep directional spectra. Time-discretization is accomplished by the TVD Runge-Kutta scheme. The overall convergence of the scheme is exponential (spectral). We successfully verified and validated the method against several analytical solutions, observational data, and experimental results. to these as source terms. The source terms have been continuously modified over the years either by including a new mechanism or improving the existing terms. These modifications over the years have been named first, second, and third generation of models; recent attempts have been towards a newer wind-wave model with an optimized source function [3, 4] . The third generation model, established by the WAMDI group  , is still today the most widely accepted model in operational ocean codes. Phase-averaged models have been widely used from large oceans to coastal regions with great success. The most well-known are from the European Center for Medium-Range Weather Forecast (ECMWF) , Simulating Waves Near-shore (SWAN) from Delft University , and NOAA's WAVEWATCH  models. Phase-resolving models, unlike phase-averaged, have diversified into many different equations based on depth characteristics, strength of nonlinearities, and bottom slopes such as Boundary Integral models; mild-slope equation models, Boussinesq models, and shallow water equations. The phase-resolving model has advantages over the pase-averaged ones in that it contains inherently nonlinear wave-interaction (triads) and better resolves the wave height in a strongly varying wave field. However, its prohibitive computation cost (much more expensive than phase-averaged models per unit area) limits it to only small regions (such as near-field of wave-structure interactions). We refer the interested readers to Battjes'  for a comprehensive review of both the phase-averaged and phase-resolving models. Application of high-order methods to phase-resolving methods (fully nonlinear water waves, Boussinesq, shallow water equations, etc.) have recently received attention with DG schemes proposed and implemented in       . In particular, phase-averaged models have adopted finite difference methods for spatial discretization in the past. Community codes such as SWAN, WAVEWATCH mix first-, second-and third-order schemes for discretization of physical and spectral space derivatives together with the first-order Euler scheme in time. Although these codes are now mature, the numerical ocean wave community is aware of the indisputable advantages of unstructured grid technology, which simplifies mesh generation and offers flexibility for controlling mesh resolution around a sharp region without increasing the grid resolution on the smooth region. Finite element and finite volume schemes using unstructured grids have recently been applied to phase-averaged models    with great success. In this paper, we present a high-order scheme in both geophysical and spectral spaces. We are not aware of any attempt to use high-order schemes on a general unstructured grid (in physical space) in the literature for phase-averaged equations. Higher-order methods can be very effective in deep ocean water simulations, which involve waves propagating over longer distances. Ocean models using a first-order scheme for the spatial advection term are limited to coastal simulations where waves propagate over relatively short distances such that the numerical solution is still not damped significantly. On the other hand, unlike geophysical space, spectral space has no tolerance for numerical diffusion of the first-order scheme; those terms are exclusively treated with second-order discretization  . Tolman  and Bender  reported improving results using second-and third-order schemes for the propagation term in physical spaces. The paper is organized as follows. We first present the action balance equation (phased-averaged equation) in Section 2, and then propose a high-order DG discretization in physical space after evaluating the frequency and directional derivatives with Fourier-collocation in Section 3. In the same section, we use a mapped Fourier-collocation for local refinement in a directional coordinate and introduce a modified action balance equation for proper treatment of the absorbing boundary condition along the frequency direction. Following this section, we present verification results to show the exponential convergence of the scheme in both physical and spectral spaces and establish the convergence order of time discretization scheme (TVD Runge-Kutta). In Section 5, we validate the new method against available analytical solutions, observational data, and experimental results. Finally, we present a brief summary in Section 6. In the appendices we include technical details on the formulation, the numerical implementation, and on the type of source terms employed, so that the interested readers can reproduce our results and follow the method more closely.