  The small energy exchange via nucleon recoils in neutrino-nucleon scattering
is now supposed to be one of the important factors for successful explosion of
core-collapse supernovae (CCSNe) as they can change neutrino spectra through
accumulation of a large number of scatterings. In finite-difference methods
employed for neutrino transport in CCSN simulations, we normally can not afford
to deploy a large enough number of energy bins needed to resolve this small
energy exchange and sub-grid techniques are employed one way or another. In
this paper we study quantitatively with the Monte Carlo (MC) method how well
such a treatment performs. We first investigate the effects of nucleon recoils
on the neutrino spectra and confirm that the average energy is reduced by
$\sim$15% for heavy-lepton neutrinos and by much smaller quantities for other
types of neutrinos in a typical post-bounce situation. It is also observed that
the nucleon scattering dominates the electron scattering in the thermalization
of neutrino spectra in all flavors. We then study possible artifacts that the
coarse energy grid may produce in the finite-difference methods. In order to
mimic the latter calculation, we re-distribute MC particles in each energy bin
after a certain interval in a couple of ways and study how the results are
affected and depend on the energy-resolution. We also discuss possible
implications of our results for the finite-difference methods.
