With the exception of the vacuum two-body problem (i.e., the coalescence of two black holes), all astrophysical systems and sources of gravitational radiation involve matter. Therefore, although the black-hole–binary problem has always been considered the paradigmatic, and more demanding, problem to solve numerically, 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 (and increasing) body of numerical investigations in the literature dealing with hydrodynamic integrations in static background spacetimes. While such a statement also holds true in the case of the MHD equations, particularly from the last few years onwards, the development still awaits thorough numerical exploration. In the purely hydrodynamic case most of the investigations 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, only began in more recent years. However, conservative approaches are turning into the most common choice for the recent code developments, not only for hydrodynamics but particularly for GRMHD (as can be inferred from Table 2).

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) still 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 but are already becoming mainstream as improved theoretical and numerical approaches, along with increasing computational power, become available. Much effort is focused on the study of neutron star instabilities, gravitational collapse, and on the coalescence of compact neutron star binaries, all scenarios being investigated with the incorporation of microphysical models for neutron star matter and magnetic fields. These three-dimensional efforts also have a vacuum counterpart, the black-hole–binary problem, which has witnessed spectacular progress in the last few years [323]. A prime driver for these investigations is the emerging possibility of soon detecting gravitational waves with the ongoing experimental endeavors. The waveform catalogues resulting from time-dependent hydrodynamic and MHD simulations provide essential help to data analysis groups, since the chances for detection are noticeably enhanced for most sources through matched-filtering techniques.

In the following, we review the status of the numerical investigations in three distinctive astrophysical scenarios, which involve strong gravitational fields and, hence, relativistic physics: gravitational collapse, accretion onto black holes, and magnetohydrodynamic evolutions 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 [18] and references therein.

5.1 Gravitational collapse

5.1.1 Core collapse supernovae

5.1.2 Black-hole formation

5.2 Accretion on to black holes

5.2.1 Disk accretion

5.2.2 Jet formation

5.2.3 Wind accretion

5.2.4 Gravitational radiation

5.3 Hydrodynamic and magnetohydrodynamic evolution of neutron stars

5.3.1 Pulsations and instabilities of relativistic stars

5.3.2 Neutron-star binary and black-hole–neutron-star coalescence

5.1.1 Core collapse supernovae

5.1.2 Black-hole formation

5.2 Accretion on to black holes

5.2.1 Disk accretion

5.2.2 Jet formation

5.2.3 Wind accretion

5.2.4 Gravitational radiation

5.3 Hydrodynamic and magnetohydrodynamic evolution of neutron stars

5.3.1 Pulsations and instabilities of relativistic stars

5.3.2 Neutron-star binary and black-hole–neutron-star coalescence

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