A powerful way to improve our understanding of the above
scenarios is through accurate, large scale, three-dimensional
numerical simulations. Nowadays, computational general
relativistic astrophysics is an increasingly important field of
research. In addition to the large amount of observational data
by high-energy X- and
-ray satellites such as Chandra, XMM-Newton, or INTEGRAL, and
the new generation of gravitational wave detectors, the rapid
increase in computing power through parallel supercomputers and
the associated advance in software technologies is making
possible large scale numerical simulations in the framework of
general relativity. However, the computational astrophysicist and
the numerical relativist face a daunting task. In the most
general case, the equations governing the dynamics of
relativistic astrophysical systems are an intricate, coupled
system of time-dependent partial differential equations,
comprising the (general) relativistic (magneto-)hydrodynamic
(MHD) 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, and so on).
Nevertheless, in some astrophysical situations of interest, e.g., accretion of matter onto compact objects or oscillations of relativistic stars, the ``test fluid'' approximation is enough to get an accurate description of the underlying dynamics. In this approximation the fluid self-gravity is neglected in comparison to the background gravitational field. This is best exemplified in accretion problems where the mass of the accreting fluid is usually much smaller than the mass of the compact object. 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.
The main purpose of this review is to summarize the existing
efforts to solve numerically the equations of (ideal) general
relativistic hydrodynamics. To this aim, the most important
numerical schemes will be presented first in some detail.
Prominence will be given to the so-called Godunov-type schemes
written in conservative form. Since [163], it has been demonstrated gradually [93
,
78
,
244
,
83,
21
,
297
,
229
] that conservative methods exploiting the hyperbolic character
of the relativistic hydrodynamic equations are optimally suited
for accurate numerical integrations, even well inside the
ultrarelativistic regime. The explicit knowledge of the
characteristic speeds (eigenvalues) of the equations, together
with the corresponding eigenvectors, provides the mathematical
(and physical) framework for such integrations, by means of
either exact or approximate Riemann solvers.
The article includes, furthermore, a comprehensive description
of ``relevant'' numerical applications in relativistic
astrophysics, including gravitational collapse, accretion onto
compact objects, and hydrodynamical evolution of neutron stars.
Numerical simulations of strong-field scenarios employing
Newtonian gravity and hydrodynamics, as well as possible
post-Newtonian extensions, have received considerable attention
in the literature and will not be covered in this review, which
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 that mimics the existence of an event
horizon at the Schwarzschild gravitational radius [217]. This has allowed accurate interpretations of observational
phenomena.
The organization of this article is as follows: Section 2 presents the equations of general relativistic hydrodynamics, summarizing the most relevant theoretical formulations that, 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 to solve nonlinear hyperbolic systems of conservation laws. Hence, particular emphasis will be paid on conservative high-resolution shock-capturing (HRSC) upwind methods based on linearized Riemann solvers. Alternative schemes such as smoothed particle hydrodynamics (SPH), (pseudo-)spectral methods, and others will be briefly discussed as well. Section 4 summarizes a comprehensive sample of hydrodynamical simulations in strong-field general relativistic astrophysics. Finally, in Section 5 we provide additional technical information needed to build up upwind HRSC schemes for the general relativistic hydrodynamics equations. Geometrized units (G = c =1) are used throughout the paper except where explicitly indicated, as well as the metric conventions of [186]. Greek (Latin) indices run from 0 to 3 (1 to 3).
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Numerical Hydrodynamics in General Relativity
José A. Font http://www.livingreviews.org/lrr-2003-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |