  We perform two-dimensional numerical simulations on the core-collapse of a
massive star with strong magnetic fields and differential rotations using a
numerical code ZEUS-2D. Changing field configurations and laws of differential
rotation parametrically, we compute 14 models and investigate effects of these
parameters on the dynamics. In our models, we do not solve the neutrino
transport and instead employ a phenomenological parametric EOS that takes into
account the neutrino emissions. As a result of the calculations, we find that
the field configuration plays a significant role in the dynamics of the core if
the initial magnetic field is large enough. Models with initially concentrated
fields produce more energetic explosions and more prolate shock waves than the
uniform field. Quadrapole-like fields produce remarkably collimated and fast
jet, which might be important for gamma-ray bursts(GRB). The Lorentz forces
exerted in the region where the plasma-beta is less than unity are responsible
for these dynamics. The pure toroidal field, on the other hand, does not lead
to any explosion or matter ejection. This suggests the presupernova models of
Heger et al.(2003), in which toroidal fields are predominant, is
disadvantageous for the magnetorotation-induced supernova considered here.
Models with initially weak magnetic fields do not lead to explosion or matter
ejection, either. In these models magnetic fields play no role as they do not
grow on the timescale considered in this paper so that the magnetic pressure
could be comparable to the matter pressure. This is because the exponential
field growth as expected in MRI is not seen in our models. The magnetic field
is amplified mainly by field-compression and field-wrapping in our simulations.
