  The proposal that core collapse supernovae are neutrino driven is still the
subject of active investigation more than fifty years after the seminal paper
by Colgate and White. The modern version of this paradigm, which we owe to
Wilson, proposes that the supernova shock wave is powered by neutrino heating,
mediated by the absorption of electron-flavor neutrinos and antineutrinos
emanating from the proto-neutron star surface, or neutrinosphere. Neutrino weak
interactions with the stellar core fluid, the theory of which is still
evolving, are flavor and energy dependent. The associated neutrino mean free
paths extend over many orders of magnitude and are never always small relative
to the stellar core radius. Thus, neutrinos are never always fluid like.
Instead, a kinetic description of them in terms of distribution functions that
determine the number density of neutrinos in the six-dimensional phase space of
position, direction, and energy, for both neutrinos and antineutrinos of each
flavor, or in terms of angular moments of these neutrino distributions that
instead provide neutrino number densities in the four-dimensional phase-space
subspace of position and energy, is needed. In turn, the computational
challenge is twofold: (i) to map the kinetic equations governing the evolution
of these distributions or moments onto discrete representations that are
stable, accurate, and, perhaps most important, respect physical laws such as
conservation of lepton number and energy and the Fermi--Dirac nature of
neutrinos and (ii) to develop efficient, supercomputer-architecture-aware
solution methods for the resultant nonlinear algebraic equations. In this
review, we present the current state of the art in attempts to meet this
challenge.
