  We discuss a new kind of astrophysical transport problem: the coherent
evolution of neutrino flavor in core collapse supernovae. Solution of this
problem requires a numerical approach which can simulate accurately the quantum
mechanical coupling of intersecting neutrino trajectories and the associated
nonlinearity which characterizes neutrino flavor conversion. We describe here
the two codes developed to attack this problem. We also describe the surprising
phenomena revealed by these numerical calculations. Chief among these is that
the nonlinearities in the problem can engineer neutrino flavor transformation
which is dramatically different than in standard Mikheyev-Smirnov-Wolfenstein
treatments. This happens even though the neutrino mass-squared differences are
measured to be small, and even when neutrino self-coupling is sub-dominant. Our
numerical work has revealed potential signatures which, if detected in the
neutrino burst from a Galactic core collapse event, could reveal heretofore
unmeasurable properties of the neutrinos, such as the mass hierarchy and vacuum
mixing angle theta_13.
