Numerical Methods for Conservation Laws

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Hyperbolic systems of conservation laws arise in many problems where theconservation of physical quantities such a mass, momentum, and energy aremodeled. These equations must typically be solved by approximate numericalmethods. Discontinuities in the data and/or solution (e.g. shock waves)cause difficulties in developing accurate and stable methods. This proposal concerns the development of high-resolution methods and theirimplementation as software that can be widely used in solving problemsthat arise in many different fields. The P.I. has been activelyinvolved in developing the CLAWPACK and AMRCLAW software formulti-dimensional hyperbolic systems. This software is freelyavailable and allows students and researchers studying a wide range ofphenomena to use the technology of high-resolution methods and adaptivemesh refinement. This software will be further developed and used toexplore a number of particular applications in science andengineering. One application concerns computing acoustic or elasticwaves in heterogeneous media. Such problems often arise in geophysicsand materials science, for example. Another application is tonumerical relativity, where there is interest in simulatinggravitational waves that may soon be detectable by astronomers.This can be accomplished by using a hyperbolic formulation of the Einsteinequations.