Numerical Methods in General Relativity

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In view of large-scale national and international programs to construct gravitational wave observatories, there is a pressing need for accurate simulation of the gravitational wave emission from massive cosmicevents such as inspiralling black hole collisions. This requires the accurate numerical simulation of the Einstein field equations, an extremely complex system of partial differential equations, which have thus far resisted attempts to develop stable numerical methods. Progress in the development of numerical methods for the Einstein equations will require both a deep physical understanding of general relativity and gravitation and the mastery of many areas of modern numerical analysis and scientific computation. The principalinvestigator will spend one year in residence at the Department of Physics of Penn State University, in a period of intense interdisciplinary collaboration and scientific interchange with a group of gravitational physicists. In this period he will complement his expertise and experience in the numerical solution of partial differential equations with the physical background and understanding necessary to develop and implement new numerical approaches to the Einstein equations. At the same time he will convey to his physicist collaborators recent advances in numerical analysis relevant to computational relativity.The interdisciplinary collaboration will focus on two directions relevant to the simulation of gravitational radiation. First, the principal investigator and his collaborators will apply recent advances in finite element technology, especially adaptive tetrahedral mesh refinement algorithms and multigrid solvers, to the generation of physically relevant initial data satisfying the constraints implied by the Einstein equations. The resulting initial data will be disseminated to the community of researchers working on numerical simulation of gravitational radiation. Second, the principal investigators and his collaborators will study the application of new adaptive unstructured mesh methods for evolution problems, such as discontinuous Galerkin methods, to the Einstein equations. These methods hold promise not only for the efficient resolution of the complex and highly variable solution structure, but also for dealing with a variety of other numerical problems that arise with the Einstein equations.This project will bring about a quantum leap in the principal investigator's involvement with numerical relativity which will continue well beyond the period of the grant. In addition to the ongoing collaboration, the principal investigator will offer a graduate course and give seminars and lectures, in order to recruit students and postdocs and to form a group in computational relativity. Through conference presentations, lectures, and articles he will work to develop awareness and interest in the numerical analysis community ofthe problems and challenges in numerical relativity. This IGMS project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).