Hyperbolic systems of partial differential equations arise in many applications where wave propagation or transport phenomena are important. Often these equations and/or their solutions involve discontinuous functions, giving difficulties for standard finite-difference approaches to discretizing the differential equations. In particular, nonlinear wave propagation problems often give rise to shock waves, discontinuities in the solution which can arise spontaneously even from smooth initial data. The goal is then to approximate a weak solution to the underlying integral conservation law. Often the problem must be solved in a heterogeneous medium where the material properties vary with space, often discontinously at sharp material interfaces. This results in discontinuous coefficients or flux functions in the equations to be solved. This proposal concerns the further development of multidimensional high-resolution finite volume methods for solving such problems, the development of software implementing these methods, and the application of these methods to particular problems. The P.I. has previously developed a multidimensional "wave-propagation algorithm" that yields a very general framework for solving such problems, and has implemented this method in the CLAWPACK software. These algorithms and the software will be further developed and brought to bear on a variety of problems. Some particular applications to be studied include: tsunami propagation and runup, pyroclastic flows arising from volcanic eruptions, the simulation of seismic waves, and elastic wave propagation in heterogeneous media, including shock wave propagation in tissue and bone.
A wide range of practical problems arising in science and engineering are modeled using "hyperbolic differential equations" and have a very similar mathematical structure, allowing researchers in applied and computational mathematics to make contributions that are widely applicable. The goal of this work is to further develop methods and software for approximating the solutions to these equations. These methods are implemented in the CLAWPACK software package written by the P.I. and co-workers, which is freely available on the web and allows students and researchers studying a wide range of phenomena to use state of the art methods for these mathematical problems. This software has been downloaded by more than 5000 registered users over the past several years and applied to numerous scientific and engineering problems by the PI, his students, and other users. Specific practical problems will also be studied, building on work already performed by the P.I. and students. One project involves modeling the effects of tsunamis on coastal regions, both to aid in scientific studies of past tsunamis and as an aid to hazard mitigation and preparedness. Other geophysical projects involve the study of flows arising from volcanic eruptions and the propagation of seismic waves in the earth following an earthquake or in oil exploration. A project with biomedical applications is the study of shock waves propagating in tissue and bone, with potential application to the study of "shock wave therapy", in which ultrasonic shock waves are used to treat a variety of medical conditions including nonunions (broken bones that fail to heal), plantar fasciitis, and tendinitis.
