We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless ͑CT͒ approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under what conditions the CT approach gives better results in 3D numerical evolutions compared to the ADM formulation. In particular, we show that our implementation of the CT approach gives more long term stable evolutions than ADM in all the cases studied, but is less accurate in the short term for the range of resolutions used in our 3D simulations.

Improvements in the performance of processors and networks make it both feasible and interesting to treat collections of workstations, servers, clusters, and supercomputers as integrated computational resources, or Grids. However, the highly heterogeneous and dynamic nature of such Grids can make application development difficult. Here we describe an architecture and prototype implementation for a Grid-enabled computational framework based on Cactus, the MPICH-G2 Grid-enabled message-passing library, and a variety of specialized features to support efficient execution in Grid environments. We have used this framework to perform record-setting computations in numerical relativity, running across four supercomputers and achieving scaling of 88% (1140 CPU's) and 63% (1500 CPUs). The problem size we were able to compute was about five times larger than any other previous run. Further, we introduce and demonstrate adaptive methods that automatically adjust computational parameters during run time, to increase dramatically the efficiency of a distributed Grid simulation, without modification of the application and without any knowledge of the underlying network connecting the distributed computers.

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