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 Wilson's Eulerian formulation of the hydrodynamic equations and use schemes based on finite differences with some amount of artificial viscosity. The use of conservative formulations of the equations, and the incorporation of the characteristic information in the design of numerical schemes has begun in more recent years.
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 interest in this direction, only the spherically symmetric case (1D) has been extensively studied. In axisymmetry (2D) fewer attempts have been made, with most of them devoted to the study of the gravitational collapse of rotating stellar cores, black hole formation, and the subsequent emission of gravitational radiation. Three-dimensional simulations have only started more recently. Much effort is nowadays focused on the study of the coalescence of compact neutron star binaries (as well as the vacuum black hole binary counterpart). These theoretical investigations are driven by the emerging possibility of soon detecting gravitational waves with the different experimental efforts currently underway. The waveform catalogues resulting from time-dependent hydrodynamical simulations may provide some help to data analysis groups, since the chances for detection may be enhanced through matched-filtering techniques.
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 neutron stars. Relativistic cosmology, another area where fundamental advances have been accomplished through numerical simulations, is not considered; the interested reader is directed to the Living Reviews article by Anninos  and references therein.
|Numerical Hydrodynamics in General Relativity
José A. Font
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