Non-linear hydrodynamical evolution of rotating relativistic stars: numerical methods and code tests

We present numerical hydrodynamical evolutions of rapidly rotating relativi stic stars, using an axisymmetric, non-linear relativistic hydrodynamics co de. We use four different high-resolution shock-capturing (HRSC) finite-dif ference schemes (based on approximate Riemann solvers) and compare their ac curacy in preserving uniformly rotating stationary initial configurations i n long-term evolutions. Among these four schemes, we find that the third-or der piecewise parabolic method scheme is superior in maintaining the initia l rotation law in long-term evolutions, especially near the surface of the star. It is further shown that HRSC schemes are suitable for the evolution of perturbed neutron stars and for the accurate identification (via Fourier transforms) of normal modes of oscillation. This is demonstrated for radia l and quadrupolar pulsations in the non-rotating limit, where we find good agreement with frequencies obtained with a linear perturbation code. The co de can be used for studying small-amplitude or non-linear pulsations of dif ferentially rotating neutron stars, while our present results serve as test bed computations for three-dimensional general-relativistic evolution codes .