The numerical calculation of three-dimensional water waves using a boundary integral method

Author: Haroldsen, David John

Year: 1997

Degree: Dissertation (Ph.D.)

Advisors: Meiron, Daniel I.; Hou, Thomas Y.

Committee Member: Unknown, Unknown

Option: Applied And Computational Mathematics

DOI: 10.7907/etyh-8140

Abstract

In this work, we consider the numerical calculation of water waves in three dimensions. One well accepted method for studying surface waves is the boundary integral method, which defines the fluid velocities at the interface in terms of integrals over the boundary of the domain in which the problem is posed. There exists a considerable body of work on the numerical study of surface waves in two dimensions. However, until recently the numerical study of surface waves was considered intractable because of the high computational cost of approximating the defining integrals.

We discuss the boundary integral formulation for the three-dimensional water wave problem and present the point vortex approximation to the singular integrals which define the particle velocities. We consider three aspects of the point vortex approximation: accuracy of the approximation, efficient means of computing solutions, and numerical stability of the scheme.

Concerning the accuracy of the point vortex method, we analyze the error associated with the approximation and show that it can be expressed as a series in odd powers of the discretization parameter h. We present quadrature rules which are highly accurate.

The efficient computation of the point vortex approximation is achieved through the use of the fast multipole algorithm, which combines long distance particle inter-actions into multipole expansions which can be efficiently evaluated. The underlying periodicity of the problem is reduced to a lattice sum which can be rapidly evaluated. We discuss the implementation of the numerical schemes in both serial and parallel computing environments.

The point vortex method is shown to be highly unstable for straightforward discretizations of the surface. We analyze the stability of the method about equilibrium and discuss methods for stabilizing the numerical schemes for both the linear and nonlinear regimes. We present numerical results which show that the method can be effectively stabilized.

In the final chapter, we present numerical results from several calculations of three-dimensional waves using the methods developed in the previous chapters.

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