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 [11] and references therein.

- 4.1 Gravitational collapse
- 4.2 Accretion onto black holes
- 4.3 Hydrodynamical evolution of neutron stars

Numerical Hydrodynamics in General Relativity
José A. Font
http://www.livingreviews.org/lrr-2003-4
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