Mathematical Biology Seminar David George University of Washington "Finite volume methods and adaptive refinement for global tsunami propagation and local inundation" 3:00 p.m., Friday, Feb 10, 2006 The shallow water equations, a system of hyperbolic conservation laws for mass and momentum, are a commonly accepted approximation governing tsunami propagation. Using these equations in their native conservative form to model tsunamis, from global propagation to local inundation, presents several challenges and requires properties not possessed by traditional numerical methods simultaneously. First, global scale tsunamis have amplitudes on the order of centimeters, meaning that tsunamis begin as a very small perturbation to a motionless body of water several kilometers deep. Global tsunami modeling, therefore, demands accurately resolving a tiny deviation from the steady-state, which arises from the nontrivial balance of hydrostatic pressure and the varying sea floor bathymetry. In the near-shore zone, tsunamis exhibit very different flow characteristics such as propagating bores and inundation of dry land, necessitating a method that can handle discontinuities and moving dry regions. We have developed shock-capturing finite volume methods for the shallow water equations that can appropriately resolve the near-shore features and inundation of tsunamis, while at the same time model global propagation accurately. The different regimes of tsunami flow belong to different spatial scales as well, and require correspondingly different grid resolutions. The long wavelength of deep ocean tsunamis (several hundred kilometers) requires a large global computing domain, yet near the shore the propagating energy is compressed and focused by bathymetry in unpredictable ways, and can lead to large variations in energy and run-up even over small localized regions. For this problem, we have used adaptive mesh refinement algorithms, allowing evolving Cartesian sub-grids that can move with the propagating waves and resolve local inundation of impacted areas in a single global scale computation. The adaptive routines are based on algorithms originally developed for gas dynamics and similar hyperbolic systems, with modifications made to deal with some difficulties exhibited by the steady-states and shorelines described above. Numerous case studies and simulations from the 2004 Indian Ocean tsunami are now being used to validate the code and some of these will be shown.