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

370 Numerical relativityThe Palma, Potsdam and Wash U groups also showed that these newformulations lead to much more stable black hole evolutions as well. Whilestandard ADM formulations can evolve black holes very accurately for a shortperiod of time, as described above, large peaks in metric functions caused by socalled‘grid-stretching’ develop instabilities, which cause the codes to crash fartoo soon to study orbits of black holes. The new formulations can significantlyextend the evolution times (by factors of two or much more) that can be achieved.In all cases, the evolutions are convergent, but seem to have larger error than thestandard ADM or Bona–Massó system. These effects were recently analysed in apaper by Alcubierre et al [52]. We are now in the process of applying these newformulations to a series of interesting spacetimes, including pure gravitationalwaves, black holes, and neutron stars, some results of which are reported below.In this and a companion paper [52] we focus on an alternative approach basedon a conformal decomposition of the metric and the trace-free components ofthe extrinsic curvature. The conformal-tracefree (CT) approach was first devisedby Nakamura in the 1980s in 3D calculations [22, 53], and then modified andapplied to work on gravitational waves [49], and on neutron stars [9, 54]. Thisapproach was not taken up by others in the community until a recent paper byBaumgarte and Shapiro [48], where a similar formulation was compared with thestandard ADM approach and shown to be superior, in terms of both accuracy andstability, on tests involving weak gravitational waves, with geodesic and harmonicslicing. In a follow-up paper, Baumgarte, Hughes, and Shapiro [55] applied thesame formulation to systems with given (analytically prescribed) matter sources,and found similar stability properties. More recently fully hydrodynamicalsimulations employing the CT approach have been reported in [56–58] in thecontext of collapse of rapidly-rotating (isolated) neutron stars and coalescenceand merge of binary neutron stars.In the companion paper [52] we perform an analytic investigation of thestability properties of the ADM and the CT evolution equations. Using alinearized plane wave analysis, we identify features of the equations that webelieve are responsible for the difference in their stability properties.18.3.0.1 Numerical techniques for the evolution equationsMost of what has been attempted in numerical relativity evolution schemes is builton explicit finite difference schemes. Implicit and iterative evolution schemeshave been occasionally attempted, but the extra cost associated has made themless popular. We now describe the basic approach that has been tried for boththe standard ADM formulation and more recent hyperbolic formulations of theequations.