Thesis (Ph.D.)--University of Hawai'i at Manoa, 2010
This dissertation describes the formulation, verification, and validation of a dispersive wave model with a shock-capturing scheme, and its implementation for basin-wide evolution and coastal runup of tsunamis using two-way nested computational grids. The depth-integrated formulation builds on the nonlinear shallow-water equations and utilizes a non-hydrostatic pressure term to describe weakly dispersive waves. The semi-implicit, finite difference solution captures flow discontinuities associated with bores or hydraulic jumps through a momentum conservation scheme, which also accounts for energy dissipation in the wave breaking process without the use of an empirical model. An upwind scheme extrapolates the free surface elevation instead of the flow depth to provide the flux in the momentum and continuity equations. This eliminates depth extrapolation errors and greatly improves the model stability, which is essential for computation of energetic breaking waves and runup
The vertical velocity term associated with non-hydrostatic pressure also describes tsunami generation and transfer of kinetic energy due to dynamic seafloor deformation. A depth-dependent Gaussian function smooths bathymetric features smaller than the water depth to improve convergence of the implicit, non-hydrostatic solution. A two-way grid-nesting scheme utilizes the Dirichlet condition of the non-hydrostatic pressure and both the velocity and surface elevation at the grid interface to ensure propagation of dispersive waves and discontinuities through computational grids of different resolution. The inter-grid boundary can adapt to topographic features to model wave transformation processes at optimal resolution and computational efficiency
The computed results show very good agreement with data from previous laboratory experiments for wave propagation, transformation, breaking, and runup over a wide range of conditions. The present model is applied to the 2009 Samoa Tsunami for demonstration and validation. These case studies confirm the validity and effectiveness of the present modeling approach for tsunami research and impact assessment. Since the numerical scheme to the momentum and continuity equations remains explicit, the implicit non-hydrostatic solution is directly applicable to existing nonlinear shallow-water models
