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 [11] and to the research papers [47,
48,
12].

- 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-2000-2
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