With the exception of the vacuum two-body problem (i.e. the coalescence of two black holes), all realistic astrophysical systems and sources of gravitational radiation involve matter. Thus not surprisingly, the joint integration of the equations of motion for matter and geometry was in the minds of theorists from the very beginning of numerical relativity.
Nowadays there is a large body of numerical investigations in the literature dealing with hydrodynamical integrations in static background spacetimes. Most of those are based on the Wilson formulation of the hydrodynamic equations and use schemes based on finite differences with some amount of artificial viscosity. In more recent years, researchers have started to use conservative formulations of the equations, and their characteristic information, in the design of numerical schemes.
On the other hand, time-dependent simulations of self-gravitating flows in general relativity, evolving the spacetime dynamically with the Einstein equations coupled to a hydrodynamic source, constitute a much smaller sample. Although there is much recent interest in this direction, only the spherically symmetric case (1D) has been extensively studied and, to some extent, can be considered essentially solved. In axisymmetry, i.e. 2D, fewer attempts have been made, with most of them devoted to the study of the gravitational collapse and bounce of rotating stellar cores and the subsequent emission of gravitational radiation. Three-dimensional simulations have only started more recently. The effort is nowadays mainly focused on the study of the coalescence and merging of compact neutron star binaries (as well as the vacuum black hole binary counterpart). These theoretical investigations are driven by the emerging possibility of detecting gravitational waves in a few years time with the different experimental efforts currently underway.
In the following we review the status of the numerical investigations in three astrophysical scenarios all involving strong gravitational fields and, hence, relativistic physics: gravitational collapse, accretion onto black holes and hydrodynamical evolution of binary neutron stars. Relativistic cosmology, another area where fundamental advances have been accomplished through numerical simulations, is not considered. The interested reader is addressed to the Living Reviews article by Anninos  and to the research papers [47, 48, 12].
|Numerical Hydrodynamics in General Relativity
José A. Font
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