  Monte Carlo approaches to radiation transport have several attractive
properties such as simplicity of implementation, high accuracy, and good
parallel scaling. Moreover, Monte Carlo methods can handle complicated
geometries and are relatively easy to extend to multiple spatial dimensions,
which makes them potentially interesting in modeling complex multi-dimensional
astrophysical phenomena such as core-collapse supernovae. The aim of this paper
is to explore Monte Carlo methods for modeling neutrino transport in
core-collapse supernovae. We generalize the Implicit Monte Carlo photon
transport scheme of Fleck & Cummings and gray discrete-diffusion scheme of
Densmore et al. to energy-, time-, and velocity-dependent neutrino transport.
Using our 1D spherically-symmetric implementation, we show that, similar to the
photon transport case, the implicit scheme enables significantly larger
timesteps compared with explicit time discretization, without sacrificing
accuracy, while the discrete-diffusion method leads to significant speed-ups at
high optical depth. Our results suggest that a combination of spectral,
velocity-dependent, Implicit Monte Carlo and discrete-diffusion Monte Carlo
methods represents a robust approach for use in neutrino transport calculations
in core-collapse supernovae. Our velocity-dependent scheme can easily be
adapted to photon transport.
