Efficient Numerical Methods for Water Wave Propagation in Unbounded Domains.

Abstract

In models of water waves, solutions are often sought on an unbounded domain. 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. To complete the description of the computational problem, boundary conditions must be prescribed on the artificial boundary. In order to effectively model the unbounded domain, the boundary conditions should make the artificial boundary invisible to outgoing waves. This type of boundary condition is often called an "absorbing" boundary condition (ABC), and they are non-trivial to formulate. The focus of this work is the derivation and numerical implementation of ABCs for the water wave equation (WWE), which describes linearized two-dimensional incompressible, irrotational, inviscid free surface flow in deep water. We derive a one-way version of the water wave equation (OWWWE), which supports the propagation of water waves essentially only in the outgoing direction. The OWWWE takes the form of a fractional partial differential equation involving a nonlocal operator corresponding to half a derivative. Properties on the one way water wave equation are given. 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. 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.