A hybrid spectral/DG method for solving the phase-averaged ocean wave equation: Algorithm and validation

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 fast 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.