A one-dimensional lattice Boltzmann model is developed to solve the shallow water equations for steady and unsteady flows within both the subcritical and supercritical regimes. Previous work is extended through a generalized Galilean transformation applied to the standard scheme. The transformation yields a general asymmetric lattice Boltzmann model scheme which can successfully model a wide range of both subcritical and supercritical flow regimes, and enables implementation of the asymmetric model for practical purposes. In current work, a new set of equilibrium functions, boundary conditions and the external force weights are derived for the generalized transformed scheme. A new stability region is also defined, allowing selection of a lattice speed that maintains numerical stability for a wider range of sub-and supercritical flows and combinations of those flow conditions, compared to the previous scheme with fixed asymmetry. The model is validated against a range of benchmark cases in open-channel hydraulics that demonstrate the applicability of the new model.

The Lattice Boltzmann Method (LBM) is a promising tool to model fluid flows. This thesis presents a summary of the investigations carried out to apply the LBM to study run-up of waves induced by bores on beaches. The thesis starts with a critical review of the common numerical models used in fluid mechanics with a specific focus on the origin and historical advances in the LBM. This indicated that at the outset of this study it was accepted that LBM application was limited to flows with subcritical regime. Hence, modelling supercritical run-up flow did not appear possible with LBM. The major achievement of current work is a one-dimensional Lattice Boltzmann Model which is developed to solve the shallow water equations for steady and unsteady flows within both the subcritical and supercritical regimes. The asymmetric LBM proposed by Chopard et al. (2013) is extended through a generalised Galilean transformation applied to the standard LBM scheme. The transformation yields a general asymmetric Lattice Boltzmann Model scheme which can successfully model a wide range of subcritical and supercritical flows, and enables implementation of the asymmetric model for practical purposes. In the current work a new set of the Equilibrium Distribution Functions, boundary conditions and the external force weights are derived for the generalised transformed scheme. A new stability region is also defined, allowing selection of a lattice speed that maintains numerical stability for a wider range of sub-and supercritical flows and combinations of those flow conditions, compared to the previous scheme with fixed asymmetry. The model is validated against a range of benchmark cases in open-channel hydraulics that demonstrate the applicability of the new model. The applicability of the model to solve nearshore problems, such as wave run-up, is studied further by a critical review of existing shoreline treatment techniques and developing a new wetting-drying boundary condition.A wetting-drying boundary condition is developed using LBM fundamentals, which is a modified version of the technique proposed by , to accommodate the transformation. The modified algorithm is successfully implemented in the transformed scheme. However, due to very shallow depths that inevitably occur in nearshore zone, the flow conditions in that area fall outside the numerical stability zone defined for the transformed scheme, resulting in instability. It is concluded that while the transformed scheme can successfully be applied to both subcritical and supercritical regimes, in its current form it has limited applicability to problems involving very shallow flows where the Froude and lattice Froude numbers will not be encompassed by the stable zone. I would like to express my sincere gratitude to my principal advisor Professor Tom Baldock for his continuous support, patience and immense knowledge. With all my work and family commitments during the course of current research work, I would not have been able to complete my PhD study without his extraordinary sereni...