A Study of Enhanced, Higher Order Boussinesq-Type Equations and Their Numerical Modelling

This project has encompassed efforts in two separate veins: on the one hand, the acquiring of highly accurate model equations of the Boussinesq-type, and on the other hand, the theoretical and practical work in implementing such equations in the form of conventional numerical models, with obvious potential for applications to the realm of numerical modelling in coastal engineering. The derivation and analysis of several forms of higher-order in dispersion and non-linearity Boussinesq-type equations have been undertaken, obtaining and investigating the properties of a new and generalised class of Boussinesq-type equations. The equations emerge from a study of arbitrarily higher-order Boussinesq-type equations in several customary choices of velocity variables. In doing so, a generalised horizontal velocity variable is defined corresponding to optimal properties, in the sense of the Padé-approximants of the fully-dispersive, fully-nonlinear starting point of the derivations. A Padé [4/4], fully-nonlinear, version of these equations is expanded on for a detailed investigation with respect to the errors intrinsic to the reduction of the dimensionality from three to two, as well as the theoretical and practical aspects of a viable and
efficient numerical solution.

Two Boussinesq-type models have been devised and tested in the course of this project. The first model is customised to the solution of higher-order Boussinesq equations, formulated in terms of the horizontal volume-flux vector. The second model is designated for the solution of higher-order Boussinesq-type equations, formulated in terms of the horizontal velocity at an arbitrary depth vector. Various discretisation techniques and grid definitions have been considered in this endeavour, undertaking a detailed analysis of the selected discretisation methods. The analysis categorises the errors of the semidiscretised and the fully-discretised equations into the categories of spurious dispersion and spurious diffusion. In particular, issues of numerical wave refraction and numerical wave blocking are introduced and addressed in the contexts of the relevant model
discretisations.

The successful application of the models to the simulation of the underlying phenomena in regards to the propagation of surface waves over a fully-submerged trapezoidal bar sheds light on the extended scope of application of such equations / models, rendering the early lower-order Boussinesq-type equations inappropriate for the simulation of the whole range of the phenomena.
The utilised higher-order equations demonstrated that with a relatively minor increase in the computational cost and algorithm complexity, a fairly major increase in the accuracy of the emulation of the phenomena is realizable. These tests also provided a venue for the practical investigation of the linear and nonlinear properties of the numerical models in the sense of the type of discretisations. Similarly, the applications to the propagation over the focusing bathymetry of Whalin (1971) was a similar venue for the assessment, in two horizontal dimensions, of the scope of the employed equations / models.