The description of important areas of modern astronomy, such as high-energy astrophysics or gravitational wave astronomy, requires general relativity. Einstein’s theory of gravitation plays a major role in astrophysical scenarios involving compact objects such as neutron stars and black holes. High-energy radiation is often emitted in regions of strong gravitational fields near such compact objects. The production of relativistic radio jets in active galactic nuclei, explained by either hydrodynamic or electromagnetic mechanisms, involves rotating supermassive black holes. The discovery of kHz quasi-periodic oscillations in low-mass X-ray binaries extended the frequency range over which these oscillations occur into timescales associated with the relativistic, innermost regions of accretion disks. A relativistic description is also necessary in scenarios involving explosive collapse of very massive stars () to black holes (in the collapsar and hypernova models), or during the last phases of the coalescence and merge of neutron star binaries and neutron-star–black-hole binaries. These catastrophic events are believed to occur at the central engine of the most highly energetic events in nature, gamma-ray bursts (GRBs). Astronomers have long been scrutinizing these systems using the complete frequency range of the electromagnetic spectrum. Nowadays, they are the main targets of ground-based laser interferometers of gravitational radiation. The direct detection of these elusive ripples in the curvature of spacetime, and the wealth of new information that could be extracted from them, is one of the driving motivations of present-day research in relativistic astrophysics.

This field is currently in a state of very rapid development. On the one hand, steady progress is being driven by observational facilities, either ground based or in space, provided by a large number of high-energy X-ray and -ray telescopes such as MAGIC and HESS, and satellites such as ASCA (which stopped operating in 2001), RXTE, Chandra, XMM-Newton, INTEGRAL, Suzaku, and GLAST (whose launch is scheduled for the first half of 2008), along with the first generation of laser-interferometer gravitational-wave detectors, which have just become operational (LIGO, VIRGO, GEO, and TAMA). On the other hand, a second factor responsible for such development is provided by the corresponding theoretical studies, which struggle to come up with models to explain those observations. Modern theoretical astrophysics has always relied on numerical simulations as a formidable way to improve our understanding of the dynamics of astrophysical systems. Presently, 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, computational astrophysicists and numerical relativists still face a daunting task. In a fairly 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 hydrodynamic and magneto-hydrodynamic (MHD) equations and the Einstein gravitational field equations. Furthermore, the number of equations must be augmented in situations, which require one to account for nonadiabatic processes, such as viscosity, resistivity, radiative transfer, or the description of the microphysics of neutron-star interiors (realistic equations of state for nuclear matter, nuclear physics, and so on).

However, in some astrophysical situations of interest, such as in studies of accretion of matter onto compact objects, the propagation of relativistic outflows and jets, or oscillations of relativistic stars, the “test-fluid” approximation is good enough to provide an accurate description of the underlying dynamics. In this approximation, the self-gravity of the fluid 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 and MHD (i.e., with the stress-energy tensor being that of a perfect fluid, which is furthermore a perfect conductor), is also a fairly standard choice in numerical astrophysics.

The scope of this review is limited to discussing the first set of equations, namely hydrodynamics and magnetohydrodynamics (GRHD and GRMHD hereafter, respectively). In particular, the various approaches and existing efforts to solve numerically those equations in general relativity will be summarized in great detail. For these two sets of equations, the mathematical framework upon which the numerical simulations are based has been developed to a high degree of sophistication, particularly for the GRHD case. To illustrate this, the most important numerical schemes will be presented to sufficient extent. Prominence will be given to the Godunov-type schemes (or upwind high-resolution shock-capturing schemes) written in conservative form. Since the early 1990s, it has gradually become manifest 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 Jacobian matrices of the system of equations, together with the corresponding eigenvectors, provides the mathematical (and physical) framework for such integrations, by means of either exact or approximate Riemann solvers.

Furthermore, this article includes a comprehensive description of relevant numerical applications in relativistic astrophysics, namely gravitational collapse, accretion onto compact objects, and evolutions 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. From the coverage of the existing numerical work presented here it becomes apparent that GRHD simulations in relativistic astrophysics are now routinely performed by an ever growing number of groups. This is particularly true within the test-fluid approximation, but also for dynamic spacetimes. Indeed, for the latter case, it is worth mentioning the long-term, numerically-stable formulations of Einstein’s equations (or accurate enough approximations) along with a number of new techniques to handle dynamic black-hole spacetimes that have been proposed by several numerical relativity groups worldwide in recent years. These have finally made possible black-hole–binary evolutions. The situation which the numerical relativist is currently confronted with, has suddenly changed for the better. Accurate and long-term stable, coupled evolutions of the GRHD/GRMHD equations and Einstein’s equations are just becoming possible in three-dimensions (benefited from the steady increase in computing power), allowing for the study of interesting relativistic astrophysical scenarios for the first time, such as those which are covered in this review: gravitational collapse, black-hole accretion, and mergers of compact binaries. While major recent advances have also been achieved for the MHD equations, the astrophysical applications investigated so far are still limited, although relevant results are already available for explaining mechanisms of jet formation. However, there can be little doubt that this field is bound to witness significant developments in the near future.

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. Correspondingly, we discuss in Section 3 the equations of general-relativistic (ideal) magnetohydrodynamics. We note that this is an entirely new section with respect to earlier versions of the review [126, 127]. Its inclusion has been motivated by the pronounced activity this topic has experienced in very recent years, as will become clear throughout the article. Section 4 is mainly devoted to describing numerical schemes specifically designed to solve nonlinear hyperbolic systems of conservation laws. Hence, particular emphasis will be paid to 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. A discussion of state-of-the-art three-dimensional GRHD/GRMHD codes is also included in Section 4. Next, Section 5 presents a comprehensive overview of relevant hydrodynamic and magnetohydrodynamic simulations in strong-field general-relativistic astrophysics. Finally, in Section 6 we provide additional technical information needed to build up upwind HRSC schemes for the general-relativistic hydrodynamics equations. Geometrized units () are used throughout the paper, except where explicitly indicated, as well as the metric conventions of [264]. The unit for the magnetic field is gauss, and the corresponding 4-vector is redefined by dividing it by a factor of . Greek (Latin) indices run from zero to three (one to three). Additional bibliography that the interested reader may find appropriate to seek advice from includes, e.g., [220, 15, 240, 217, 417, 41, 323, 158].

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