  Accurate neutrino transport has been built into spherically symmetric
simulations of stellar core collapse and postbounce evolution. The results of
such simulations agree that spherically symmetric models with standard
microphysical input fail to explode by the delayed, neutrino-driven mechanism.
Independent groups implemented fundamentally different numerical methods to
tackle the Boltzmann neutrino transport equation. Here we present a direct and
detailed comparison of such neutrino radiation-hydrodynamical simulations for
two codes, Agile-Boltztran of the Oak Ridge-Basel group and Vertex of the
Garching group. The former solves the Boltzmann equation directly by an
implicit, general relativistic discrete angle method on the adaptive grid of a
conservative implicit hydrodynamics code with second-order TVD advection. In
contrast, the latter couples a variable Eddington factor technique with an
explicit, moving-grid, conservative high-order Riemann solver with important
relativistic effects treated by an effective gravitational potential. The
presented study is meant to test both neutrino radiation-hydrodynamics
implementations and to provide a data basis for comparisons and verifications
of supernova codes to be developed in the future. Results are discussed for
simulations of the core collapse and post-bounce evolution of a 13 solar mass
star with Newtonian gravity and a 15 solar mass star with relativistic gravity.
