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We present our first successful numerical results of 3D general relativisticsimulations in which the Einstein equation as well as the hydrodynamicequations are fully solved. This paper is especially devoted to simulations oftest problems such as spherical dust collapse, stability test of perturbedspherical stars, and preservation of (approximate) equilibrium states ofrapidly rotating neutron star and/or corotating binary neutron stars. Thesetest simulations confirm that simulations of coalescing binary neutron starsare feasible in a numerical relativity code. It is illustrated that using ournumerical code, simulations of these problems, in particular those ofcorotating binary neutron stars, can be performed stably and fairly accuratelyfor a couple of dynamical timescales. These numerical results indicate that ourformulation for solving the Einstein field equation and hydrodynamic equationsare robust and make it possible to perform a realistic simulation of coalescingbinary neutron stars for a long time from the innermost circular orbit up toformation of a black hole or neutron star.Comment: 36 pages, to be published in PRD 15, erase unnecessary figure

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