The Shallow Water Equations with Topography Equations for Depth and Momentum

Tsunamis belong to a class of geophysical problems with vastly different spatial scales of interest. For instance, the wave propagation of a global scale tsunami may result in wave run-up and inundation that varies greatly even along local stretches of coastline. This diversity of scales presents a difficulty to numerical modelers—the accuracy that is desired at a local scale requires a grid resolu-tion that is simply not feasible to use at the global scale. Additionally, waves that propagate throughout the global domain may be concentrated in localized areas at a given time. Therefore, using fixed telescoping grids is not always a satisfactory solution. We have used adaptive mesh refinement algorithms, originally developed for gas dynamics, to numerically model global scale tsunamis. These algorithms allow regions with grids of varying refinement where the solution has steep gradients or other features of interest. The regions of refinement may move adaptively with the solution over time, allowing various resolutions in a single global scale computation