The explosion of massive stars involves the formation of a shock wave. In stars that develop iron cores, this shock wave stalls on its way out due to neutrino emission and the breakup of heavy nuclei flowing through the shock. For the explosion to succeed, a fraction of the gravitational binding energy of the collapsed core that is radiated in neutrinos needs to be absorbed by the material below the shock. How much energy is needed depends on the interplay between non-spherical hydrodynamic instabilities, neutrino heating, and nuclear dissociation. This thesis seeks to understand this interplay through numerical experiments that model the key physical components of the system and separate them out to examine their individual effects. Specifically, one- and two dimensional time-dependent hydrodynamic simulations are performed to study the effects of non-spherical shock oscillations, neutrino-driven convection, and alpha particle recombination on the dynamics of the system and the critical heating rate for explosion.

We find that nuclear dissociation has a significant effect on the linear stability and saturation amplitude of shock oscillations. At the critical neutrino heating rate for an explosion, convection due to a negative entropy gradient plays a major role in driving dipolar shock motions. One dimensional explosions are due to a global instability involving the advection of entropy perturbations from the shock to the region where the accretion flow cools due to neutrino emission. Large scale shock expansions in two-dimensions are due to a finite amplitude instability involving the balance between buoyancy forces and the ram pressure of the flow upstream of the shock. During these expansions, a significant amount of energy is released when nucleons recombine into alpha particles, constituting a significant last step in the transition to explosion. The critical neutrino heating rate for an explosion depends sensitively on the starting radius of the shock relative to the radius at which the binding energy of an alpha particle is comparable to the gravitational binding energy.