A powerful way to improve our understanding of such events is through accurate, large scale, three-dimensional numerical simulations. In the most general case, the equations governing the dynamics of those systems are an intricate, coupled system of time-dependent partial differential equations, comprising the (general) relativistic (magneto-) hydrodynamic equations and the Einstein gravitational field equations. In many cases, the number of equations must be augmented to account for non-adiabatic processes, e.g. radiative transfer or sophisticated microphysics (realistic equations of state for nuclear matter, nuclear physics, magnetic fields, etc.).

Nevertheless, in some astrophysical situations of interest,
e.g. accretion of matter onto compact objects, the `test-fluid'
approximation is commonly adopted and suffices to get an accurate
description. In this approximation the fluid self-gravity is
neglected in comparison to the
*background*
gravitational field, the core assumption being that the mass
of the accreting fluid is much smaller than the mass of the
compact object
*M*, i.e.
. For instance, a black hole accreting matter at the Eddington
rate
would need about
years to double its mass, which certainly justifies the
assumption. Additionally, a description employing ideal
hydrodynamics (i.e. with the stress-energy tensor being that of a
perfect fluid), is also a fairly standard choice in numerical
astrophysics.

It is the main aim of this review to summarize the existing efforts to solve the equations of (ideal) general relativistic hydrodynamics by numerical means. For this purpose, the most relevant numerical schemes will be presented in some detail. Furthermore, relevant applications in a number of different astrophysical systems, including gravitational collapse, accretion onto compact objects and hydrodynamical evolution of neutron stars, will also be summarized here.

Numerical simulations of strong-field scenarios employing Newtonian gravity and hydrodynamics, as well as possible post-Newtonian extensions, which have received considerable attention in the literature, will not be covered, since this review focuses on relativistic simulations. Nevertheless, we must emphasize that most of what is known about hydrodynamics near compact objects, in particular in black hole astrophysics, has been accurately described using Newtonian models. Probably the best known example is the use of a pseudo-Newtonian potential for non-rotating black holes which mimics the existence of an event horizon at the Schwarzschild gravitational radius [167], which has allowed accurate interpretations of observational phenomena.

The organization of this paper is as follows: Section 2 presents the equations of general relativistic hydrodynamics, summarizing the most relevant theoretical formulations which, to some extent, have helped to drive the development of numerical algorithms for their solution. Section 3 is mainly devoted to describing numerical schemes specifically designed for non-linear hyperbolic systems. Hence, particular emphasis will be paid on conservative high-resolution shock-capturing methods based on linearized Riemann solvers. Also alternative schemes such as smoothed particle hydrodynamics (SPH) and (pseudo-) spectral methods will be briefly discussed. Section 4 summarizes a comprehensive sample of hydrodynamical simulations in strong-field general relativistic astrophysics.

Geometrized units (*G*
=
*c*
=1) are used throughout the paper except where explicitly
indicated, as well as the metric conventions of [143]. Greek (Latin) indices run from 0 to 3 (1 to 3).

Numerical Hydrodynamics in General Relativity
José A. Font
http://www.livingreviews.org/lrr-2000-2
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