Modelling of tidal flows in coastal regions

Abstract

In this thesis a combination of high-order central finite difference techniques and composite overlapping grid me thods are used for solving the two dimensional shallow water equations (SWE) in a tidal flow modelling context. 1\vo main topics are discussed in this dissertation. First, an energy-stable high-order central finite difference scheme is derived for the SWE. This scheme is based on the semi-discrete energy method for initial boundary value problems for symmetrisable nonlinear hyperbolic conservation laws, described by Olsson (1 995). After symmetrising the SWE, via a change to entropy variables, the flux derivatives are then split using a technique called entropy-splitting. This is the main step which allows a continuous energy estimate for the SWE to be derived. A high-order central finite difference operator which satisfies summation-by-parts is then introduced, enabling the formulation of a semi-discrete energy estimate. A new exact travelling vortex solution of the SWE is given. Following the treatment for the Euler equations of gas dynamics in Yee (2000), this exact solution is used to verify experimentally that the entropy-splitting improves the stability properties of the fully discrete equations. The numerical scheme based on these methods is referred to here as the Energy Stable High-Order scheme (ESHO). Second, the composite overlapping grid method (COGM) is investigated by carrying out a series of numerical experiments which solve the two dimensional shallow water equations in conjunction with the ESHO scheme. The first numerical experiments test the performance of a selection of Lagrangian interpolation methods, in conjunction with the ESHO scheme and various composite overlapping grids, to solve the travelling vortex problem. The fourth order implicit Lagrangian interpolation scheme is found to have the best performance in terms of grid convergence. To illustrate the advantages of the combined COGM/ESHO scheme (incorporating the chosen interpolation scheme) it is applied to several idealised test problems that are representative of tidal flows in coastal regions. A composite overlapping grid design strategy is used which is appropriate for these test problems, however the numerical method may be used in combination with any valid composite overlapping grid. Further work is suggested for extending the methods proposed here to simulating tidal flows in realistic spatial domains.