  We have been working within the fundamental paradigm that core collapse
supernovae (CCSNe) may be neutrino driven, since the first suggestion of this
by Colgate and White nearly five decades ago. Computational models have become
increasingly sophisticated, first in one spatial dimension assuming spherical
symmetry, then in two spatial dimensions assuming axisymmetry, and now in three
spatial dimensions with no imposed symmetries. The increase in the number of
spatial dimensions has been accompanied by an increase in the physics included
in the models, and an increase in the sophistication with which this physics
has been modeled. Computation has played an essential role in the development
of CCSN theory, not simply for the obvious reason that such multidimensional,
multi-physics, nonlinear events cannot possibly be fully captured analytically,
but for its role in discovery. In particular, the discovery of the standing
accretion shock instability (SASI) through computation about a decade ago has
impacted all simulations performed since then. Today, we appear to be at a
threshold, where neutrinos, neutrino-driven convection, and the SASI, working
together over time scales significantly longer than had been anticipated in the
past, are able to generate explosions, and in some cases, robust explosions, in
a number of axisymmetric models. But how will this play out in three
dimensions? Early results from the first three-dimensional (3D), multi-physics
simulation of the "Oak Ridge" group are promising. I will discuss the essential
components of today's models and the requirements of realistic CCSN modeling,
present results from our one-, two-, and three-dimensional models, place our
models in context with respect to other efforts around the world, and discuss
short- and long-term next steps.
