  We present an implicit finite difference representation for general
relativistic radiation hydrodynamics in spherical symmetry. Our code,
Agile-Boltztran, solves the Boltzmann transport equation for the angular and
spectral neutrino distribution functions in self-consistent simulations of
stellar core collapse and postbounce evolution. It implements a dynamically
adaptive grid in comoving coordinates. Most macroscopically interesting
physical quantities are defined by expectation values of the distribution
function. We optimize the finite differencing of the microscopic transport
equation for a consistent evolution of important expectation values. We test
our code in simulations launched from progenitor stars with 13 solar masses and
40 solar masses. ~0.5 s after core collapse and bounce, the protoneutron star
in the latter case reaches its maximum mass and collapses further to form a
black hole. When the hydrostatic gravitational contraction sets in, we find a
transient increase in electron flavor neutrino luminosities due to a change in
the accretion rate. The muon- and tauon-neutrino luminosities and rms energies,
however, continue to rise because previously shock-heated material with a
non-degenerate electron gas starts to replace the cool degenerate material at
their production site. We demonstrate this by supplementing the concept of
neutrinospheres with a more detailed statistical description of the origin of
escaping neutrinos. We compare the evolution of the 13 solar mass progenitor
star to simulations with the MGFLD approximation, based on a recently developed
flux limiter. We find similar results in the postbounce phase and validate this
MGFLD approach for the spherically symmetric case with standard input physics.
