  One- and two-dimensional hydrodynamical simulations of neutrino-driven
supernova explosions are discussed. The simulations cover the phase between the
stagnation of the prompt shock and about one second after core bounce.
Systematic variation of the neutrino fluxes from the neutrino sphere shows that
the explosion energy, explosion time scale, initial mass of the protoneutron
star, and explosive nucleosynthesis of iron-group elements depend sensitively
on the strength of the neutrino heating during the first few 100 ms after shock
formation. Convective overturn in the neutrino-heated region behind the shock
is a crucial help for the explosion only in a narrow window of neutrino
luminosities. Here powerful explosions can be obtained only in the
multi-dimensional case. For higher core-neutrino fluxes also spherically
symmetrical models yield energetic explosions, while for lower luminosities
even with convection no strong explosions occur.
