Nonlinear hydrodynamical evolution of rotating relativistic stars: Numerical methods and code tests

TL;DR

This paper develops and tests numerical methods for simulating the evolution of rotating relativistic stars, demonstrating the effectiveness of high-resolution shock-capturing schemes, especially the third-order PPM scheme, in maintaining initial configurations and analyzing stellar oscillations.

Contribution

The paper introduces a comparative analysis of four HRSC schemes for relativistic star simulations, highlighting the superior performance of the third-order PPM scheme in long-term evolution accuracy.

Findings

PPM scheme best preserves initial rotation law

HRSC schemes accurately identify stellar oscillation modes

Good agreement with linear perturbation frequencies

Abstract

We present numerical hydrodynamical evolutions of rapidly rotating relativistic stars, using an axisymmetric, nonlinear relativistic hydrodynamics code. We use four different high-resolution shock-capturing (HRSC) finite-difference schemes (based on approximate Riemann solvers) and compare their accuracy in preserving uniformly rotating stationary initial configurations in long-term evolutions. Among these four schemes, we find that the third-order PPM scheme is superior in maintaining the initial 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 radial and quadrupolar pulsations in the nonrotating limit, where we find good agreement with frequencies obtained with a linear perturbation code. The code can be used for studying small-amplitude or nonlinear pulsations of differentially rotating neutron stars, while our present results serve as testbed computations for three-dimensional general-relativistic evolution codes.