Three-dimensional numerical general relativistic hydrodynamics. II. Long-term dynamics of single relativistic stars

This is the second in a series of papers on the construction and validation of a three-dimensional code for the solution of the coupled system of the Einstein equations and of the general relativistic hydrodynamic equations, and on the application of this code to problems in general relativistic astrophysics. In particular, we report on the accuracy of our code in the long-term dynamical evolution of relativistic stars and on some new physics results obtained in the process of code testing. The following aspects of our code have been validated: the generation of initial data representing perturbed general relativistic polytropic models (both rotating and nonrotating), the long-term evolution of relativistic stellar models, and the coupling of our evolution code to analysis modules providing, for instance, the detection of apparent horizons or the extraction of gravitational waveforms. The tests involve single nonrotating stars in stable equilibrium, nonrotating stars undergoing radial and quadrupolar oscillations, nonrotating stars on the unstable branch of the equilibrium configurations migrating to the stable branch, nonrotating stars undergoing gravitational collapse to a black hole, and rapidly rotating stars in stable equilibrium and undergoing quasiradial oscillations. We have carried out evolutions in full general relativity and compared the results to those obtained either with perturbation techniques, or with lower dimensional numerical codes, or in the Cowling approximation (in which all the perturbations of the spacetime are neglected). In all cases an excellent agreement has been found. The numerical evolutions have been carried out using different types of polytropic equations of state using either the rest-mass density only, or the rest-mass density and the internal energy as independent variables. New variants of the spacetime evolution and new high resolution shock capturing treatments based on Riemann solvers and slope limiters have been implemented and the results compared with those obtained from previous methods. In particular, we have found the ``monotonized central differencing'' limiter to be particularly effective in evolving the relativistic stellar models considered. Finally, we have obtained the first eigenfrequencies of rotating stars in full general relativity and rapid rotation. A long standing problem, such frequencies have not been obtained by other methods. Overall, and to the best of our knowledge, the results presented in this paper represent the most accurate long-term three-dimensional evolutions of relativistic stars available to date.