Finite Volume Methods for Hyperbolic Problems

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Hyperbolic systems of partial differential equations arise in manyapplications where wave propagation or transport phenomena are important.Often these equations and/or their solutions involve discontinuousfunctions, giving difficulties for standard finite-difference approachesto discretizing the differential equations. In particular, nonlinear wavepropagation problems often give rise to shock waves, discontinuities inthe solution which can arise spontaneously even from smooth initial data.The goal is then to approximate a weak solution to the underlying integralconservation law. Often the problem must be solved in a heterogeneousmedium where the material properties vary with space, often discontinouslyat sharp material interfaces. This results in discontinuous coefficientsor flux functions in the equations to be solved. This proposal concernsthe further development of multidimensional high-resolutionfinite volume methods for solving such problems, the development ofsoftware implementing these methods, and the application of thesemethods to particular problems. The P.I. has previously developeda multidimensional "wave-propagation algorithm" that yields a verygeneral framework for solving such problems, and has implemented thismethod in the CLAWPACK software. These algorithms and the softwarewill be further developed and brought to bear on a variety of problems.Some particular applications to be studied include: tsunami propagationand runup, pyroclastic flows arising from volcanic eruptions, thesimulation of seismic waves, and elastic wave propagation in heterogeneousmedia, including shock wave propagation in tissue and bone.A wide range of practical problems arising in science and engineering aremodeled using "hyperbolic differential equations" and have a very similarmathematical structure, allowing researchers in applied and computationalmathematics to make contributions that are widely applicable. The goalof this work is to further develop methods and software for approximatingthe solutions to these equations. These methods are implemented in theCLAWPACK software package written by the P.I. and co-workers, whichis freely available on the web and allows students and researchersstudying a wide range of phenomena to use state of the art methods forthese mathematical problems. This software has been downloaded by morethan 5000 registered users over the past several years and applied tonumerous scientific and engineering problems by the PI, his students, andother users. Specific practical problems will also be studied, building onwork already performed by the P.I. and students. One project involvesmodeling the effects of tsunamis on coastal regions, both to aid inscientific studies of past tsunamis and as an aid to hazard mitigationand preparedness. Other geophysical projects involve the study of flowsarising from volcanic eruptions and the propagation of seismic waves inthe earth following an earthquake or in oil exploration. A project withbiomedical applications is the study of shock waves propagating in tissueand bone, with potential application to the study of "shock wave therapy",in which ultrasonic shock waves are used to treat a variety of medicalconditions including nonunions (broken bones that fail to heal), plantarfasciitis, and tendinitis.