We present a new formulation of the Einstein equations that casts them in an explicitly rst order, ux-conservative, hyperbolic form. We show that this now can be done for a wide class of time slicing conditions, including maximal slicing, making it potentially very useful for numerical relativity. This development permits the application to the Einstein equations of advanced numerical methods developed to solve the uid dynamic equations, without overly restricting the time slicing, for the rst time. The full set of characteristic elds and speeds is explicitly given.PACS numbers: 04.25.Dm Introduction. For decades the standard approach 1] to numerical relativity has been based on a direct application of the 3+1 formulation of the Einstein equations by Arnowitt, Deser, and Misner 2]. This important contribution laid the foundation for most numerical work in the eld. As convenient as this formulation is for numerical relativity, the structure of the equations is extremely complicated and most of the work consisted in developing \ad hoc" numerical codes for every case considered. This is in contrast with the situation of modern Computational Fluid Dynamics, which deals with rst order, ux conservative, hyperbolic (FOFCH) systems of equations, for which many advanced numerical methods have been developed 3] based on the particular mathematical structure of the equations 4].After many pioneering attempts 5{7] it was shown that the full set of 3D Einstein equations could be put in the FOFCH form 8], similar to hydrodynamics. This development allowed the same advanced numerical methods used in hydrodynamics, such as conservative schemes and modern shock capturing methods, to be applied to the Einstein equations for the rst time. In a number of tests involving spherical black holes 9,10], 1D general relativistic hydrodynamics 11] and 3D gravitational waves 12], this formulation showed some of its strengths over the standard approach. However, the price to pay for this original formulation was that it required the restrictive assumption of harmonic time slicing. Although this slicing condition is singularity avoiding, making it potentially useful for studies of strongly gravitating systems, it is just barely so 13]. For this reason metric and curvature components can grow without bound near the singularity and, without some sort of horizon boundary condition 10,14,15], this slicing is not well suited for black hole spacetimes.In this Letter, we will show that this FOFCH formal-
Abstract. We describe Cactus, a framework for building a variety of computing applications in science and engineering, including astrophysics, relativity and chemical engineering. We first motivate by example the need for such frameworks to support multi-platform, high performance applications across diverse communities. We then describe the design of the latest release of Cactus (Version 4.0) a complete rewrite of earlier versions, which enables highly modular, multi-language, parallel applications to be developed by single researchers and large collaborations alike. Making extensive use of abstractions, we detail how we are able to provide the latest advances in computational science, such as interchangeable parallel data distribution and high performance IO layers, while hiding most details of the underlying computational libraries from the application developer. We survey how Cactus 4.0 is being used by various application communities, and describe how it will also enable these applications to run on the computational Grids of the near future. Application Frameworks in Scientific ComputingVirtually all areas of science and engineering, as well as an increasing number of other fields, are turning to computational science to provide crucial tools to further their disciplines. The increasing power of computers offers unprecedented ability to solve complex equations, simulate natural and man-made complex processes, and visualise data, as well as providing novel possibilities such as new forms of art and entertainment. As computational power advances rapidly, computational tools, libraries, and computing paradigms themselves also advance. In such an environment, even experienced computational scientists and engineers can easily find themselves falling behind the pace of change, while they redesign and rework their codes to support the next computer architecture. This
We report on a new 3D numerical code designed to solve the Einstein equations for general vacuum spacetimes. This code is based on the standard 3+1 approach using Cartesian coordinates. We discuss the numerical techniques used in developing this code, and its performance on massively parallel and vector supercomputers. As a test case, we present evolutions for the first 3D black hole spacetimes. We identify a number of difficulties in evolving 3D black holes and suggest approaches to overcome them. We show how special treatment of the conformal factor can lead to more accurate evolution, and discuss techniques we developed to handle black hole spacetimes in the absence of symmetries. Many different slicing conditions are tested, including geodesic, maximal, and various algebraic conditions on the lapse. With current resolutions, limited by computer memory sizes, we show that with certain lapse conditions we can evolve the black hole to about t = 50M , where M is the black hole mass. Comparisons are made with results obtained by evolving spherical initial black hole data sets with a 1D spherically symmetric code. We also demonstrate that an "apparent horizon locking shift" can be used to prevent the development of large gradients in the metric functions that result from singularity avoiding time slicings. We compute the mass of the apparent horizon in these spacetimes, and find that in many cases it can be conserved to within about 5% throughout the evolution with our techniques and current 1 resolution.
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