This thesis describes a finite-volume BGK kinetic scheme with MUSCL-type methodology based on the numerical discretization of the approximate Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collisional model for the purpose of solving numerically the time-dependent non-linear non-homogeneous shallow-water equations in one and two space dimensions with source terms arising from bottom topography. The intrinsic connection between the BGK kinetic model with the inclusion of the gravitational force term and the shallow-water equations with source terms due to the varying bed profile enables us to automatically solve the shallow-water equations from the BGK scheme. Being a well-balanced method to replicate the stationary flow, the BGK scheme employed the surface gradient method (SGM)...[ Read more ]

This thesis describes a finite-volume BGK kinetic scheme with MUSCL-type methodology based on the numerical discretization of the approximate Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collisional model for the purpose of solving numerically the time-dependent non-linear non-homogeneous shallow-water equations in one and two space dimensions with source terms arising from bottom topography. The intrinsic connection between the BGK kinetic model with the inclusion of the gravitational force term and the shallow-water equations with source terms due to the varying bed profile enables us to automatically solve the shallow-water equations from the BGK scheme. Being a well-balanced method to replicate the stationary flow, the BGK scheme employed the surface gradient method (SGM) [27] in the initial data reconstruction stage, as well as implemented with the particle velocity change due to the gravitational force and the varying river channel bed in the explicit flux evaluation to preserve the steady states given by the flow at rest initially. Problems associated with dam-break modelling, shallow-water flow of a shock around a circular cylinder, stationary and quasi-stationary flows, roll-waves down an inclined open channel, and waves incident on a slope beach will be presented, and accurate and robust numerical approximations are obtained.