  We perform a series of two-dimensional magnetohydrodynamic simulations of the
rotational core-collapse of a magnetized massive star. We employ a realistic
equation of state and take into account the neutrino cooling by the so-called
leakage scheme. In this study we systematically investigate how the strong
magnetic field and the rapid rotation affect the propagation of the shock
waves. Our results show that in the case of the strong initial poloidal
magnetic field, the toroidal magnetic field amplified by the differential
rotation, becomes strong enough to generate a tightly collimated shock wave
along the rotational axis. On the other hand, in the case of the weak initial
magnetic field, although the differential rotation amplifies toroidal magnetic
field over the long rotational period, the launched shock wave is weak and the
shape of it becomes wider. The former case is expected to be accompanied by the
formation of the so-called magnetar. Our models with rapid rotation and strong
magnetic field can create a nozzle formed by the collimated shock wave. This
might be the analogous situation of the collapsar that is plausible for the
central engine of the Gamma-Ray Bursts.
