Local characteristic algorithms for relativistic hydrodynamics

Numerical schemes for the general relativistic hydrodynamic equations are discussed. The use of conservative algorithms based upon the characteristic structure of those equations, developed during the last decade building on ideas first applied in Newtonian hydrodynamics, provides a robust methodology to obtain stable and accurate solutions even in the presence of discontinuities. The knowledge of the wave structure of the above system is essential in the construction of the so-called linearized Riemann solvers, a class of numerical schemes specifically designed to solve nonlinear hyperbolic systems of conservation laws. In the last part of the review some astrophysical applications of such schemes, using the coupled system of the (characteristic) Einstein and hydrodynamic equations, are also briefly presented.Comment: 20 pages, 4 figures, To appear in the proceedings of the workshop "The conformal structure of space-time", J. Frauendiener, H. Friedrich, eds, Springer Lecture Notes in Physic

Non-axisymmetric relativistic Bondi-Hoyle accretion onto a Schwarzschild black hole

We present the results of an exhaustive numerical study of fully relativistic non-axisymmetric Bondi-Hoyle accretion onto a moving Schwarzschild black hole. We have solved the equations of general relativistic hydrodynamics with a high-resolution shock-capturing numerical scheme based on a linearized Riemann solver. The numerical code was previously used to study axisymmetric flow configurations past a Schwarzschild hole. We have analyzed and discussed the flow morphology for a sample of asymptotically high Mach number models. The results of this work reveal that initially asymptotic uniform flows always accrete onto the hole in a stationary way which closely resembles the previous axisymmetric patterns. This is in contrast with some Newtonian numerical studies where violent flip-flop instabilities were found. As discussed in the text, the reason can be found in the initial conditions used in the relativistic regime, as they can not exactly duplicate the previous Newtonian setups where the instability appeared. The dependence of the final solution with the inner boundary condition as well as with the grid resolution has also been studied. Finally, we have computed the accretion rates of mass and linear and angular momentum.Comment: 21 pages, 13 figures, Latex, MNRAS (in press

The runaway instability of thick discs around black holes. I. The constant angular momentum case

We present results from a numerical study of the runaway instability of thick discs around black holes. This instability is an important issue for most models of cosmic gamma-ray bursts, where the central engine responsible for the initial energy release is such a system consisting of a thick disc surrounding a black hole. We have carried out a comprehensive number of time-dependent simulations aimed at exploring the appearance of the instability. Our study has been performed using a fully relativistic hydrodynamics code. The general relativistic hydrodynamic equations are formulated as a hyperbolic flux-conservative system and solved using a suitable Godunov-type scheme. We build a series of constant angular momentum discs around a Schwarzschild black hole. Furthermore, the self-gravity of the disc is neglected and the evolution of the central black hole is assumed to be that of a sequence of exact Schwarzschild black holes of varying mass. The black hole mass increase is thus determined by the mass accretion rate across the event horizon. In agreement with previous studies based on stationary models, we find that by allowing the mass of the black hole to grow the disc becomes unstable. Our hydrodynamical simulations show that for all disc-to-hole mass ratios considered (between 1 and 0.05), the runaway instability appears very fast on a dynamical timescale of a few orbital periods, typically a few 10 ms and never exceeding 1 s for our particular choice of the mass of the black hole ($2.5 \mathrm{M_\odot}$) and a large range of mass fluxes (\dot{m} \ga 10^{-3} \mathrm{M_{\odot}/s}). The implications of our results in the context of gamma-ray bursts are briefly discussed.Comment: 20 pages, 16 figures, to appear in MNRA

"Mariage des Maillages": A new numerical approach for 3D relativistic core collapse simulations

We present a new 3D general relativistic hydrodynamics code for simulations of stellar core collapse to a neutron star, as well as pulsations and instabilities of rotating relativistic stars. It uses spectral methods for solving the metric equations, assuming the conformal flatness approximation for the three-metric. The matter equations are solved by high-resolution shock-capturing schemes. We demonstrate that the combination of a finite difference grid and a spectral grid can be successfully accomplished. This "Mariage des Maillages" (French for grid wedding) approach results in high accuracy of the metric solver and allows for fully 3D applications using computationally affordable resources, and ensures long term numerical stability of the evolution. We compare our new approach to two other, finite difference based, methods to solve the metric equations. A variety of tests in 2D and 3D is presented, involving highly perturbed neutron star spacetimes and (axisymmetric) stellar core collapse, demonstrating the ability to handle spacetimes with and without symmetries in strong gravity. These tests are also employed to assess gravitational waveform extraction, which is based on the quadrupole formula.Comment: 29 pages, 16 figures; added more information about convergence tests and grid setu

Dynamics of magnetized relativistic tori oscillating around black holes

We present a numerical study of the dynamics of magnetized, relativistic, non-self-gravitating, axisymmetric tori orbiting in the background spacetimes of Schwarzschild and Kerr black holes. The initial models have a constant specific angular momentum and are built with a non-zero toroidal magnetic field component, for which equilibrium configurations have recently been obtained. In this work we extend our previous investigations which dealt with purely hydrodynamical thick discs, and study the dynamics of magnetized tori subject to perturbations which, for the values of the magnetic field strength considered here, trigger quasi-periodic oscillations lasting for tens of orbital periods. Overall, we have found that the dynamics of the magnetized tori analyzed is very similar to that found in the corresponding unmagnetized models. The spectral distribution of the eigenfrequencies of oscillation shows the presence of a fundamental p mode and of a series of overtones in a harmonic ratio 2:3:.... These simulations, therefore, extend the validity of the model of Rezzolla et al.(2003a) for explaining the high-frequency QPOs observed in the spectra of LMXBs containing a black-hole candidate also to the case of magnetized discs with purely toroidal magnetic field distribution. If sufficiently compact and massive, these oscillations can also lead to the emission of intense gravitational radiation which is potentially detectable for sources within the Galaxy.Comment: 10 pages,7 figures, submitted to MNRA

Gravitational waves from oscillating accretion tori: Comparison between different approaches

Quasi-periodic oscillations of high density thick accretion disks orbiting a Schwarzschild black hole have been recently addressed as interesting sources of gravitational waves. The aim of this paper is to compare the gravitational waveforms emitted from these sources when computed using (variations of) the standard quadrupole formula and gauge-invariant metric perturbation theory. To this goal we evolve representative disk models using an existing general relativistic hydrodynamics code which has been previously employed in investigations of such astrophysical systems. Two are the main results of this work: First, for stable and marginally stable disks, no excitation of the black hole quasi-normal modes is found. Secondly, we provide a simple, relativistic modification of the Newtonian quadrupole formula which, in certain regimes, yields excellent agreement with the perturbative approach. This holds true as long as back-scattering of GWs is negligible. Otherwise, any functional form of the quadrupole formula yields systematic errors of the order of 10%.Comment: 6 pages and 3 figures, RevTex, accepted for publication in Phys. Rev.