  We introduce a new discrete-ordinate scheme for solving the general
relativistic Boltzmann transport equation in the context of core-collapse
supernovae. Our algorithm avoids the need to spell out the complicated
advection terms in energy and angle that arise when the transport equation is
formulated in spherical polar coordinates, in the comoving frame, or in a
general relativistic spacetime. We instead approach the problem by calculating
the advection of neutrinos across momentum space using an intuitive
particle-like approach that has excellent conservation properties and fully
accounts for Lorentz boosts, general relativistic effects, and grid geometry
terms. In order to avoid the need for a global implicit solution, time
integration is performed using a locally implicit Lax-Wendroff scheme that
correctly reproduces the diffusion limit. This will facilitate the use of our
method on massively parallel distributed-memory architectures. We have verified
the accuracy and stability of our scheme with a suite of test problems in
spherical symmetry and axisymmetry. To demonstrate that the new algorithm works
stably in core-collapse supernova simulations, we have coupled it to the
general relativistic hydrodynamics code CoCoNuT and present a first
demonstration run of a $20M_\odot$ progenitor with a reduced set of neutrino
opacities.
