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

378 Numerical relativityeach cell interface. This implies that there are no Riemann solutions involved(either exact or approximate); moreover, the scheme has been proved to alleviateseveral numerical pathologies associated to the introduction of an averaged state(as Roe’s method does) in the local diagonalization procedure (see [92, 97]).For a detailed comparison of the three schemes and their coupling to dynamicalevolution of spacetimes, see [95].The availability of the hyperbolic hydrotreatment and its coupling to thespacetime evolution code is particularly noteworthy. With the development ofa hyperbolic formulation of the Einstein equations described above, the entiresystem can be treated as a single system of hyperbolic equations, rather thanartificially separating the spacetime part from the fluid part.18.3.2 Boundary conditionsAppropriate conditions for the outer boundary have yet to be derived for 3Dnumerical relativity. In 1D and 2D relativity codes, the outer boundary isgenerally placed far enough away that the spacetime is nearly flat there, and staticor flat (i.e. copying data from the next-to-last zone to the outer edge) boundaryconditions can usually be specified for the evolved functions. However, due tothe constraints placed on us by limited computer memory, this is not currentlypossible in 3D. Adaptive mesh refinement will be of great use in this regard, butwill not substitute for proper physical treatment. Most results to date have beencomputed with the evolved functions kept static at the outer boundary, even if theboundaries are too close for comfort in 3D!There are several other approaches under development that promise toimprove this situation greatly that we will not have room to explore in detail here,but should be mentioned. Generally, one has in mind using Cauchy evolution inthe strong field, interior region where, say, black holes are colliding. The outerpart of this region will be matched to some exterior treatment designed to handlewhat is primarily expected to be outgoing radiation.Two major approaches have been developed by the NSF Black Hole GrandChallenge Alliance, a large USA collaboration working to solve the black holecoalescence problem, and other groups. First, by using perturbation theory, it ispossible to identify quantities in the numerically evolved metric functions thatobey the Regge–Wheeler and Zerilli wave equations that describe gravitationalwaves propagating on a black hole background. These can be used to provideboundary conditions on the metric and extrinsic curvature functions in an actualevolution, as described in a recent paper [100]. This is an excellent stepforward in outer boundary treatments that should work to minimize reflectionsof the outgoing wave signals from the outer boundary. In tests with weakwaves, a full 3D Cauchy evolution code has been successfully matched to theperturbative treatment at the boundary, permitting waves to escape from theinterior region with very little reflection. Alternatively, ‘Cauchy-characteristicmatching’ attempts to match spacelike slices in the Cauchy region to null slices