  We present a covariant ray tracing algorithm for computing high-resolution
neutrino distributions in general relativistic numerical spacetimes with
hydrodynamical sources. Our formulation treats the very important effect of
elastic scattering of neutrinos off of nuclei and nucleons (changing the
neutrino's direction but not energy) by incorporating estimates of the
background neutrino fields. Background fields provide information about the
spectra and intensities of the neutrinos scattered into each ray. These
background fields may be taken from a low-order moment simulation or be
ignored, in which case the method reduces to a standard state-of-the-art ray
tracing formulation. The method handles radiation in regimes spanning optically
thick to optically thin. We test the new code, highlight its strengths and
weaknesses, and apply it to a simulation of a neutron star merger to compute
neutrino fluxes and spectra, and to demonstrate a neutrino flavor oscillation
calculation. In that environment, we find qualitatively different fluxes,
spectra, and oscillation behaviors when elastic scattering is included.
