Date of Award

Spring 1-1-2013

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Applied Mathematics

First Advisor

Bengt Fornberg

Second Advisor

Harvey Segur

Third Advisor

Natasha Flyer

Abstract

A Hamiltonian formulation of surface water wave dynamics offers several useful features for numerical simulations of coastal waves, including reduction of the fully three-dimensional fluid problem to free surface variables and conservation of an approximated total energy. The spectral linear version of the Hamiltonian dynamical equations captures wavelength dispersion using a pseudodifferential operator, while higher-order approximations of the total energy lead to dynamical equations that incorporate nonlinear effects in terms of this operator. These models, derived for constant depth, are extended for use with varying-depth bathymetries by replacing the pseudodifferential operator with a symmetrized combination of several such operators evaluated at selected depths from the bathymetry at hand. This new operator is constructed so as to minimize its error from the true local-depth operator over the entire bathymetry, with priority given to wavelengths for which the most accurate modeling is desired. The resulting equations are implemented using a Fourier pseudospectral method, with damping regions used to manage the inherent periodic boundary conditions, source terms used to generate waves within the computational domain, and additional damping terms used to roughly simulate reflective interfaces. The implementation is validated by comparison to data from several benchmark experiments: a focusing wave group over uniform depth, irregular waves over a sloping bathymetry, and monochromatic waves over a challenging shoal bathymetry. The results demonstrate the promising ability of this approach to accurately simulate dispersive and bathymetric effects, and to achieve improved accuracy through the use of nonlinear terms. These improvements in accuracy, however, appear to be limited by the increasing degree of filtration of wavenumber modes required to control aliasing in models of increasing order. Finally, the implementation is demonstrated through simulations of realistic wave scenarios over actual coastal bathymetries.

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