Efficient Numerical Methods for Water Wave Propagation in Unbounded Domains

In models of water waves, it is often convenient to seek solutions on an unbounded domain. For example, when modeling flow around a vessel on an open sea, effects from the distant ocean bottom or shorelines are negligible compared to the surface wave-body interactions. In order to generate computer simulations of such models, the finiteness of computer memory demands that the domain must first be truncated to finite size. This is done by introducing an artificial boundary at an arbitrary distance from the region of interest, ideally not too far away in order to minimize computational costs. To complete the description of the computational problem, boundary conditions need to be prescribed on the artificial boundary. To effectively model the unbounded domain, the boundary conditions should make the artificial boundary invisible to outgoing waves, so that waves leaving the computational domain act as if the boundary did not exist at all. This type of boundary condition is often referred to in the literature as an "absorbing" or "non-reflecting" boundary condition, and they are non-trivial to formulate.;The focus of this work is the derivation and numerical implementation of absorbing boundary conditions for the water wave equation (WWE), which describes linearized two-dimensional incompressible irrotational free surface flow. We derive a one-way version of the water wave equation, which supports the propagation of water waves essentially only in the outgoing direction. The one-way water wave equation (OWWWE) lakes the form of a fractional partial differential equation involving a nonlocal operator corresponding to half a derivative. The fractional derivative is implemented as a derivative of a convolution with a singular kernel with locally integrable singularity. Properties on the one way water wave equation are given. We solve the full water wave equation in the interior of the domain arid the one way equation in an absorbing layer near the outer artificial boundary. Additional damping may be incorporated.;We develop a hierarchy of efficient numerical methods of increasing order for numerically simulating solutions to the OWWWEs. We view the OWWWE as a conservation law with linear nonlocal flux, and use solution cell averages to compute a conservative polynomial reconstruction of the solution in each computational cell. The flux at cell interfaces is computed by evaluating exactly the convolution integral of the approximating polynomial interpolants, in what we call an Exact Polynomial Integration Computation (EPIC). Time integration uses Runge-Kutta schemes of matching order. We analyze the stability of the resulting schemes, study the convergence of the numerical solution, and present numerical results.;For the absorbing boundary, we solve the WWE in a central domain but use the OWWWEs in layers near the artificial boundary. As waves leave the central domain, they are picked up by the OWWWEs and propagated outward toward the edge of the computational domain. Numerical results are presented, In addition, techniques for damping the equations are provided along with numerical examples.