Fully general relativistic simulation of coalescing binary neutron stars: Preparatory tests

We present our first successful numerical results of 3D general relativistic simulations in which the Einstein equation and the hydrodynamic equations are fully solved. This paper is especially devoted to simulations of test problems such as spherical dust collapse, stability test of perturbed spherical stars, and preservation of (approximate) equilibrium states of rapidly rotating neutron star and/or corotating binary neutron stars. These test simulations confirm that simulations of coalescing binary neutron stars are feasible in a numerical relativity code. It is illustrated that using our numerical code, simulations of these problems, in particular those of corotating binary neutron stars, can be performed stably and fairly accurately for a couple of dynamical time scales. These numerical results indicate that our formulation for solving the Einstein field equation and hydrodynamic equations is robust and makes it possible to perform a realistic simulation of coalescing binary neutron stars for a long time from the innermost circular orbit up to formation of a black hole or neutron star.