  We present numerical results on three-dimensional (3D) hydrodynamic
core-collapse simulations of an $11.2 M_{\odot}$ star. By comparing one-(1D)
and two-dimensional(2D) results with those of 3D, we study how the increasing
spacial multi-dimensionality affects the postbounce supernova dynamics. The
calculations were performed with an energy-dependent treatment of the neutrino
transport that is solved by the isotropic diffusion source approximation
scheme. By performing a tracer-particle analysis, we show that the maximum
residency time of material in the gain region is shown to be longer for 3D due
to non-axisymmetric flow motions than 2D, which is one of advantageous aspects
of 3D models to obtain neutrino-driven explosions. Our results show that
convective matter motions below the gain radius become much more violent in 3D
than 2D, making the neutrino luminosity larger for 3D. Nevertheless the emitted
neutrino energies are made smaller due to the enhanced cooling. Our results
indicate whether these advantages for driving 3D explosions could or could not
overwhelm the disadvantages is sensitive to the employed numerical resolutions.
An encouraging finding is that the shock expansion tends to become more
energetic for models with finer resolutions. To draw a robust conclusion, 3D
simulations with much more higher numerical resolutions and also with more
advanced treatment of neutrino transport as well as of gravity is needed, which
could be hopefully practicable by utilizing forthcoming Petaflops-class
supercomputers.
