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The proposal that core collapse supernovae are neutrino driven is still thesubject of active investigation more than fifty years after the seminal paperby Colgate and White. The modern version of this paradigm, which we owe toWilson, proposes that the supernova shock wave is powered by neutrino heating,mediated by the absorption of electron-flavor neutrinos and antineutrinosemanating from the proto-neutron star surface, or neutrinosphere. Neutrino weakinteractions with the stellar core fluid, the theory of which is stillevolving, are flavor and energy dependent. The associated neutrino mean freepaths extend over many orders of magnitude and are never always small relativeto the stellar core radius. Thus, neutrinos are never always fluid like.Instead, a kinetic description of them in terms of distribution functions thatdetermine the number density of neutrinos in the six-dimensional phase space ofposition, direction, and energy, for both neutrinos and antineutrinos of eachflavor, or in terms of angular moments of these neutrino distributions thatinstead provide neutrino number densities in the four-dimensional phase-spacesubspace of position and energy, is needed. In turn, the computationalchallenge is twofold: (i) to map the kinetic equations governing the evolutionof these distributions or moments onto discrete representations that arestable, accurate, and, perhaps most important, respect physical laws such asconservation of lepton number and energy and the Fermi--Dirac nature ofneutrinos and (ii) to develop efficient, supercomputer-architecture-awaresolution methods for the resultant nonlinear algebraic equations. In thisreview, we present the current state of the art in attempts to meet thischallenge.

Physical, numerical, and computational challenges of modeling neutrino transport in core-collapse supernovae