Abstract: 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 recangular 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.
