A new implementation for magnetohydrodynamics (MHD) simulations in full general relativity (involving dynamical spacetimes) is presented. In our implementation, Einstein's evolution equations are evolved by a BSSN formalism, MHD equations by a high-resolution central scheme, and induction equation by a constraint transport method. We perform numerical simulations for standard test problems in relativistic MHD, including special relativistic magnetized shocks, general relativistic magnetized Bondi flow in stationary spacetime, and a longterm evolution for self-gravitating system composed of a neutron star and a magnetized disk in full general relativity. In the final test, we illustrate that our implementation can follow winding-up of the magnetic field lines of magnetized and differentially rotating accretion disks around a compact object until saturation, after which magnetically driven wind and angular momentum transport inside the disk turn on.04.25. Dm, 04.40.Nr, 47.75.+f, 95.30.Qd

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We perform fully general relativistic simulations of rotating stellar core collapse in three spatial dimensions. The hydrodynamic equations are solved using a high-resolution shock-capturing scheme. A parametric equation of state is adopted to model collapsing stellar cores and neutron stars following Dimmelmeier et al. The early stage of the collapse is followed by an axisymmetric code. When the stellar core becomes compact enough, we start a three-dimensional simulation adding a bar-mode nonaxisymmetric density perturbation. The axisymmetric simulations are performed for a wide variety of initial conditions changing the rotational velocity profile, parameters of the equations of state, and the total mass. It is clarified that the maximum density, the maximum value of the compactness, and the maximum value of the ratio of the kinetic energy T to the gravitational potential energy W ( T=W) achieved during the stellar collapse and bounce depend sensitively on the velocity profile and the total mass of the initial core and equations of state. It is also found that for all the models with a high degree of differential rotation, a funnel structure is formed around the rotational axis after the formation of neutron stars. For selected models in which the maximum value of is larger than 0:27, three-dimensional numerical simulations are performed. It is found that the bar-mode dynamical instability sets in for the case that the following conditions are satisfied: (i) the progenitor of the stellar core collapse should be rapidly rotating with the initial value of 0:01 & & 0:02, (ii) the degree of differential rotation for the velocity profile of the initial condition should be sufficiently high, and (iii) a depletion factor of pressure in an early stage of collapse should be large enough to induce a significant contraction to form a compact stellar core for which an efficient spin-up can be achieved surmounting the strong centrifugal force. As a result of the onset of the bar-mode dynamical instabilities, the amplitude of gravitational waves can be by a factor of 10 larger than that in the axisymmetric collapse. It is found that a dynamical instability with the m 1 mode is also induced for the dynamically unstable cases against the bar mode, but the perturbation does not grow significantly and, hence, it does not contribute to an outstanding amplification of gravitational waves. No evidence for fragmentation of the protoneutron stars is found in the first few 10 msec after the bounce.