Discrete and continuous Hamiltonian systems for wave modelling

The main focus of this thesis is to develop numerical discretisations for both compressible and incompressible inviscid flows that also preserve conservation laws at the discrete level. Two alternative approaches are discussed in detail: a semi-analytical solution; and, a fully numerical discretisation. The semi-analytical solution is derived for the case of incompressible inertial gyroscopic waves in a three-dimensional rotating rectangular parallelepiped. By performing a detailed numerical comparison it is shown that the semi-analytical solution vastly improves on the state-of-the-art solution previously available in the literature. Despite the improved accuracy of this new semi-analytical solution, further comprehensive investigations revealed a small weakness near the boundaries of the domain. A novel finite element method is derived that compensates for the known weaknesses in both the original and new semi-analytical approaches. This numerical approach allowed further investigation of inertial waves and associated physical phenomena. Subsequently, the thesis investigates the development of conservation law preserving Hamiltonian discretisations modelling (in)compressible fluid flow in three-dimensional domains with various boundary conditions. The continuous Hamiltonian description of the physical phenomenon is discretised via a discontinuous Galerkin method, which allows the construction of highly stable, conservative, energy preserving numerical discretisations for both the compressible and incompressible cases. This numerical scheme preserves the Hamiltonian mathematical structure even at the discrete level, which facilitates highly accurate and robust simulations of (in)compressible fluid flows. For the particular case of inertial gyroscopic waves this numerical scheme is more robust and accurate than the corresponding aforementioned semi-analytical solutions. Finally, a new version (version 2) of the in-house open-source C{\verb:++:} software that enables fast and easy implementation of discontinuous Galerkin discretisations (hpGEM) is introduced. The discussion evolves around the philosophy, design principles and aim of the package. Additionally, a set of new features and guidelines how to use the package are highlighted via a series of illustrative, small, step-by-step examples.