Fourier pseudospectral methods for 2D Boussinesq-type equations

► A high-order method to solve model equations for dispersive waves is presented. ► The model is compared to approximate analytical solutions and low-order solutions. ► The model is shown to be applicable in 2D simulations of wave propagation. A global Fourier pseudospectral method is presented and...

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Bibliographic Details
Published in:Ocean modelling (Oxford) 2012-08, Vol.52-53, p.76-89
Main Authors: Steinmoeller, D.T., Stastna, M., Lamb, K.G.
Format: Article
Language:English
Subjects:
Boussinesq equations
Filtering
Fluid dynamics
Fourier analysis
Hydrostatic pressure
Marine
Mathematical analysis
Mathematical models
Shallow water
Shallow water equations
Water waves
Wave dispersion
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Summary:► A high-order method to solve model equations for dispersive waves is presented. ► The model is compared to approximate analytical solutions and low-order solutions. ► The model is shown to be applicable in 2D simulations of wave propagation. 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.
ISSN:1463-5003
1463-5011
DOI:10.1016/j.ocemod.2012.05.003