(ed.). Gravitational waves (IOP, 2001)(422s).

Einstein equations for relativity 363the solution. Because of this richness and complexity of the equations,and the interesting applications to problems such as black holes and neutronstars, natural collaborations have developed between applied mathematicians,physicists, astrophysicists and computational scientists in the development of asingle code to attack these problems. There are various large-scale collaborativeefforts in recent years in this direction, including the NSF Black Hole GrandChallenge Project (recently concluded), the NASA Neutron Star Grand ChallengeProject, the NCSA/Potsdam/Wash U numerical relativity collaboration, and mostrecently a large European collaboration of ten institutions funded by the EU [5].We will describe the Cactus Computational Toolkit, along with some ofits algorithms and capabilities of this code, and a number of its applications toproblems of black holes, gravitational waves, and neutron stars. In the nextsections we will first give a brief description of the numerical formulation ofthe theory of general relativity, and discuss particular difficulties associated withnumerical relativity. The discussion will necessarily be brief. Examples aremostly drawn from work carried out by the NCSA/Potsdam/Wash U numericalrelativity collaboration. We also provide URL addresses for web pages containinggraphics and movies of some of our results.To conclude this brief introduction, a statement of where we stand in termsof simulating general relativistic compact objects is in order. The NSF black holegrand challenge project and related work achieved long term stable evolution ofsingle black hole spacetimes under certain conditions [6–8], but there is still along way to go before the spiralling coalescence can be computed. The presentlyon-going NASA neutron star grand challenge project recently succeeded inevolving grazing collision of two neutron stars using the full Einstein-relativistichydrodynamic system of equations, with a simple equation of state, and theJapanese groups also report preliminary success in evolving several orbits witha fully relativistic GR-hydro code [9]. The recently funded EU Network [5] willcontinue on the momentum of these projects. However, the final goal of a fullsolution of the problem including radiation transport and magnetohydrodynamicsfor comparison between numerical simulations and observations in gravitationalwaveastronomy (waveform templates) and high-energy astronomy (γ -ray bursts)will take many more years, hopefully building on the effort described in thispaper. So although much work has been done, much more remains, and newcommunity tools, including Cactus, are being made available to all groups. Weare very optimistic about the future, and hope to see more involvement in thiseffort across the relativity and astrophysics communities.18.2 Einstein equations for relativityThe generality and complexity of the Einstein equations make them an excellentand fertile testing ground for a variety of broadly significant computing issues.They form a system of dozens of coupled, nonlinear equations, with thousands of