Fourier pseudospectral methods for 2D Boussinesq-type equations

A global Fourier pseudospectral method is presented and used to solve a dispersive model of shallow water wave motions. The model equations under consideration are from the Boussinesq hierarchy of equations, and allow for appropriate modeling of dispersive short-wave phenomena by including weakly non-hydrostatic corrections to the hydrostatic pressure in the shallow water model. A numerical solution procedure for the Fourier method is discussed and analyzed in some detail, including details on how to efficiently solve the required linear systems. Two time-stepping approaches are discussed. Sample model results are presented, and the Fourier method is compared to the discontinuous Galerkin finite element method (DG-FEM) at various orders of accuracy. The present work suggests that scalable Fourier transform methods can be employed in water-wave problems involving variable bathymetry and can also be an effective tool at solving elliptic problems with variable coefficients if combined properly with iterative linear solvers and pre-conditioning. Additionally, we demonstrate: (1) that the small amounts of artificial dissipation (from filtering) inherent to the Fourier method make it a prime candidate for hypothesis-testing against water wave field data, and (2) the method may also serve as a benchmark for lower order numerical methods (e.g., Finite Volume Method, DG-FEM) that can be employed in more general geometries.