We discuss the high performance computing issues involved in the numerical simulation of binary neutron star mergers and supernovae. These phenomena, which are of great interest to astronomers and physicists, can only be described by modeling the gravitational field of the objects along with the flow of matter and radiation in a self consistent manner. In turn, such models require the solution of the gravitational field equations, Eulerian hydrodynamic equations, and radiation transport equations. This necessitates the use of scalable, high performance computing assets to conduct the simulations. We discuss some of the parallel computing aspects of this challenging task in this paper.