Hyperbolic systems of partial differential equations often arise when modeling phenomena involving wave propagation or advective flow. Finite volume methods are a natural approach for conservation laws of this form since they are based directly on integral formulations and are applicable to problems involving shock waves and other discontinuities. High-resolution shock-capturing methods developed originally for compressible gas dynamics can also be applied to many other hyperbolic systems. A general formulation of these methods has been developed in the CLAWPACK software that allows application of these methods, with adaptive mesh refinement, to a variety of problems in fluid and solid dynamics.

I will describe these methods in the context of some recent work on modeling geophysical flow problems, particularly in the study of tsunamis. Accurate prediction of their propagation through the ocean and interaction with coastal topography is essential in issuing early warnings and in the study of historical tsunamis. Modeling wave motion at the shore is complicated by the fact that grid cells change between wet and dry as the wave moves in and out. Special Riemann solvers have been developed to deal with dry states in order to capture the shoreline location on a rectangular grid. Propagation of small amplitude waves over deep ocean when the bathymetry varies on much larger scales than the wave amplitude will also cause numerical problems unless the method is properly formulated. Adaptive mesh refinement is desirable in order to allow much greater resolution near the shore than in the open ocean, but introduces new difficulties with varying bathymetry and dry cells. I will describe some recent progress and joint work with David George and Marsha Berger.