PhD Defense: Enhanced Fully Nonlinear Boussinesq-type Equations in Conserved Variable Form and Linear Analytical Properties with Compact Finite Difference Schemes

Troy W. Heitmann
PhD Candidate
Department of Ocean and Resources Engineering
University of Hawai’i at Mānoa

**This defense will be held both in person (POST 723) and over Zoom**
Meeting ID: 974 2551 3125
Passcode: ORE
https://hawaii.zoom.us/j/97425513125

In coastal engineering applications, Boussinesq-type models are limited by orders of approximation originating in both the governing equations and numerical schemes employed. Dispersive model solutions reflect a composition of approximations dependent upon finite sampling intervals. This study aims to improve understanding of both theoretical and numerical facets, with the end goal of strengthening practitioner awareness in model applicability.

A modern approach to parameterize wave breaking in Boussinesq-type equations is to leverage the hyperbolic structure of the leading order nonlinear shallow water equations and approximate over-turning processes using shock capturing methods designed to conserve both mass and momentum. In this approach, it is well known that the governing partial differential equations (PDEs) must be expressed in conserved variable form to attain proper shock speeds. A new independent formulation covering a family of fully nonlinear, weakly dispersive Boussinesq-type equations is derived in conserved variable form by depth integrating Euler’s equations of motion under an irrotational flow assumption. A projected Taylor series expansion of the vertical velocity about an arbitrary material surface is utilized in the depth integration of the irrotational flow condition to give an expression for the horizontal velocity. Through a change of variables, the dependency of the horizontal velocity is expressed with reference to an arbitrary point of evaluation. A weighted average of horizontal velocity expansions at the material surfaces defines the model velocity at a datum invariant reference attached to the flow depth. In comparison to existing theories, the approach introduces an additional term which enhances nonlinear dispersion. Imposing constraints on the orders of approximation, leading order theories are recovered, thus showing theoretical advancement.

Transforming the governing PDEs into discrete approximations facilitates numerical simulation of nonlinear processes over a complex bathymetry, in which the approximations result in a system of modified PDEs (MPDEs) possessing unique solutions specific to the numerical methods employed. In practical application, practitioners are burdened with an unnecessary level of uncertainty during the selection of discretization parameters, despite their fundamental roles in the governing MPDEs. Beyond numerical experiments, there has been little effort to explicitly communicate numerical implications in Boussinesq-type models. For the Boussinesq-type equations derived herein, dispersion emerges through Taylor series expansions along the vertical axis, in which the methods of approximation mirror those used in finite difference methods. Therefore, a complementary finite difference framework is adopted in which the time integration is performed using linear multistep schemes. Difference operators, including those with compact support, are expressed in symbolic form for the purpose of generalization. Applying Fourier-Laplace transforms, the symbolic operators are mapped into spectral space, where waveform resolution is evaluated as a function of time, Δt, and space, Δx, sampling intervals. The approach facilitates a complex propagation factor analysis of amplitude and phase modulations, both of which may be present in physical theory. To better accommodate operator interactions occurring in systems of equations, definitions of operator support and coefficients are adjourned to maintain complex degrees of freedom during the full system analysis. As a result, the solution to the MPDEs becomes an objective, as opposed to an outcome, when defining schemes.

Developments on Boussinesq-type equations have largely focused on dispersion enhancements to the governing PDEs, in which family members having the same formal order of accuracy exhibit very different dispersive behaviors. The same level of research has not been carried out with regard to the respective MPDEs, where different schemes lead to unique dispersive solutions. The linearized MPDEs are cast into spectral space using Fourier-Laplace transforms. Substituting in the symbolic operators, the dispersion relation matches that of the PDEs provided the discrete operators are replaced by their continuous counterparts. The dispersion relation of the MPDEs is dependent upon not only on the wave number, k, and still water depth, h, but also Δt and Δx sampling intervals. The function domain of the celerity, or phase speed, is thus multidimensional, collapsing only to k and h in the limit of vanishing sampling intervals for stable consistent schemes. Several leading order Boussinesq-type equations are analyzed, in which the error associated with the MPDEs quantifies the bounds for application. Theories which exhibit increases in phase speed with relative depth are best suited to finite difference methods. This is due to a counter balancing decrease in phase speed imposed by finite difference methods. The transparency of error associated with the MPDEs gives practitioners further insights on the selection of sampling intervals and permits optimal mesh design for a given application.