Optimization of variational Boussinesq models

The research presented in this thesis is concentrated mostly around the properties of the Variational Boussinesq Model (VBM). The VBM is a model for waves above a layer of ideal fluid, which conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate dispersion characteristics. In particular, dispersive properties of a model are important for a large number of practical applications, especially if modelled waves have a broad spectrum. The most important assessment for a model is a comparison between results obtained from the model and experimental data. We show how well the VBM performs in a few complex cases, such as focussing wave groups with broad spectra. The real-life experiments with this type of waves were performed at the facilities of the MARIN hydrodynamic laboratory (Maritime Research Institute Netherlands). In order to put the work into the context of existing research, we compare the obtained results to those of other models, in particular the AB-equation which is briefly discussed in the thesis as well. Having free parameters in the VBM gives opportunities to optimize the parameters depending on the specifcs of the application. We explore possibilities of such optimization, interchanging norms in different optimization criteria. Then the question rises, which of these norms is the best? We come up with the novel kinetic energy optimization criterium that is natural for the VBM and gives seemingly the best result in the considered test cases. Another important property from the practical viewpoint is how well a model simulates reflection. We study the reflective properties of the VBM and compare them to previous results by other authors. We also derive and investigate an analytical reflection model of the WKB (Wentzel–Kramers–Brillouin) type, whose performance is surprisingly good. For the numerical implementation of the VBM we employ a Finite Element Method (FEM), and a pseudo-spectral method in case of the AB-equation. In the thesis we concentrate on errors caused by the modelling process, and provide details of the numerical implementation in the first chapters and in the Appendix. This includes an embedded influx condition, which we use in signalling problems for both the AB-equation and VBM.