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Richard Matzner, James C. Browne, University of Texas; Larry Smarr, H. Ed Seidel, Paul Saylor, Faisal Saied, University of Illinois; Geoffrey Fox, Syracuse University; Stuart Shapiro, Saul Teukolsky, Cornell University; James York, Charles Evans, University of North Carolina; L. Samuel Finn, Northwestern University; Pablo Laguna, Penn State University; Jeffrey Winicour, University of Pittsburgh

The object of the Binary Black Holes Grand Challenge Alliance is to produce a catalog of the gravitational wave signatures from the strong gravitational field of orbiting astrophysical binary black holes and from the merger and coalescence of these bodies. This work will be carried out by a computational solution of Einstein's equations describing gravitational fields. This is a large computational problem because of the complexity of the equations, the need for a full three- dimensional simulation for data with no special symmetries, and the fact that analysis predicts a range of spatial and temporal scales from the phenomena of the process.

The project group is interested in black holes as sources for gravitational radiation because they are the most compact objects and have the strongest gravitational fields theoretically possible. The basic concept of a black hole can be explained by examining the concept of escape velocity, which is determined by the minimum energy needed to overcome the gravity of an astrophysical body and fly off into space. For instance, the escape velocity from the earth is 11.2 km/sec. If mass were added to the earth (without changing its radius), the escape velocity would be greater. To get a black hole the size of the earth, one would have to fit about 2000 times the mass of the sun into a sphere of the earth's radius. The escape velocity would then exceed the velocity of light, and light or anything else for that matter, would not escape from the surface.

Although the concepts are simple, the implementation is intricate. Einstein described gravity as a warping of space and time (spacetime); to find consistent solutions for the gravitational configuration requires the solution of a complicated set of coupled elliptical and hyperbolic equations. Furthermore, black hole spacetime phenomena occur on a succession of scales. The smallest scale arises by defining the hole radius as unity. Then, another scale is the wavelength associated with ring-down oscillations of the black hole, about 20 times as large. Adequate extraction of the radiation waveforms requires an extraction zone of hundreds of times the hole radius, setting a large outer scale for the problem.

The nested structure thus implies the need for large grids (~(103)3) and for adaptive refinements to handle changing scales as the black holes orbit through the grid. Large computers are needed to run these problems. Therefore, the group is developing parallel codes on machines including the Cray T3D at the Pittsburgh Supercomputing Center and the Thinking Machines CM-5 at the National Center for Supercomputing Applications (Illinois), as well as on architectures like the Pittsburgh Cray C90 and the Cray Y-MP at the University of Texas at Austin.

The surface of a black hole is a causal boundary, analogous to a supersonic point in fluid mechanics. Signals can propagate in only one direction across it. Thus it is sensible and beneficial to consider differencing that computes only the outside domain of the problem, which can radiate to infinity. Because the locations of the causal boundary depend on the (computational) solutions obtained, it is clear that some sort of iterative procedure will be needed to roughly estimate the gravitational field, roughly locate the causal boundaries, and then refine.

The study of the gravitational field can be cast in one of two equivalent ways that differ greatly in the way they describe the gravitational field. One way treats the gravitational field as a configuration with given derivatives that is evolved by hyperbolic equations into the future of the problem. The second method treats the three-dimensional characteristics of the problem, which are the loci of the signal flow from events in the central region out to infinity, and evolves the system from one to the next of these characteristic hypersurfaces. The characteristic approach has great a priori utility in getting radiation to propagate to infinity, but there is much more experience in handling the space-evolving-in-time approach (called the Cauchy approach) in the central strong field region. Techniques are needed for joining two substantially different descriptions at some boundary surrounding the central, dynamical part of the simulation.

Managing such a complicated problem among a number of research centers dictates a structured computational approach. With Browne and Matt Choptuik (a research scientist at the University of Texas), CRPC researcher Geoffrey Fox has begun developing the computational backbone for this project. At a meeting on May 6 (in Pittsburgh), the co-PIs of the project adopted a parallel data-structure standard. Fox has set up a framework with a software librarian for code modules meeting the standard. A closed Mosaic repository is being established that will allow PIs to check out modules for development. An initial toolkit and backbone will be deposited to the librarian at Syracuse and be made available via Mosaic.

Current work is proceeding in Fortran 77 and Fortran 90, and development work for High Performance Fortran (HPF) is being carried out to assure that HPF provides the compiler and run-time support appropriate to this problem.

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