Finite volume methods and adaptive refinement for tsunami propagation and inundation

The shallow water equations are a commonly accepted governing system for tsunami propagation and inundation. In their most generally valid form, the equations are a set of hyperbolic integral conservation laws---a general class of systems for which an extensive body of numerical theory exists. In this thesis, finite volume wave propagation methods---high resolution Godunov-type methods---are extended to this form of the shallow water equations in the context of tsunami modeling. A novel approximate Riemann solver is developed in order to handle the diverse flow regimes exhibited by tsunamis. This solver provides well-balanced source term inclusion required for accurate resolution of near steady state solutions---a necessity when modeling transoceanic tsunami propagation. The solver also preserves nonnegative water depths and accurately captures discontinuities and moving shorelines, making it appropriate for inundation modeling. Adaptive refinement algorithms are extended to this application. These algorithms allow evolving sub-grids of various resolutions to move with features in the solution. Extending the adaptive algorithms to tsunami modeling requires some new interpolation and integrating strategies in order to preserve steady states. Finally, the methods are extended to solution on a sphere or idealized earth-fitted reference ellipsoids. Together, the methods developed allow modeling transoceanic tsunami propagation as well as coastal inundation in single global-scale simulations.