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

  • Authors:

  • B. Yildirim;George Em Karniadakis

  • Affiliations:

  • School of Engineering, Brown University, Providence, RI 02912, USA;Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2012

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Abstract

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.