  We implement a multi-group and discrete-ordinate neutrino transport model in
spherical symmetry which allows to simulate collective neutrino oscillations by
including realistic collisional rates in a self-consistent way. We utilize this
innovative model, based on strategic parameter rescaling, to study a recently
proposed collisional flavor instability caused by the asymmetry of emission and
absorption rates between $\nu_e$ and $\bar\nu_e$ for four different static
backgrounds taken from different stages in a core-collapse supernova
simulation. Our results confirm that collisional instabilities generally exist
around the neutrinosphere during the SN accretion and post-accretion phase, as
suggested by [arXiv:2104.11369]. However, the growth and transport of flavor
instabilities can only be fully captured by models with global simulations as
done in this work. With minimal ingredient to trigger collisional
instabilities, we find that the flavor oscillations and transport mainly affect
(anti)neutrinos of heavy lepton flavors around their decoupling sphere, which
then leave imprints on their energy spectra in the free-streaming regime. For
electron (anti)neutrinos, their properties remain nearly intact. We also
explore various effects due to the decoherence from neutrino-nucleon
scattering, artificially enhanced decoherence from emission and absorption,
neutrino vacuum mixing, and inhomogeneous matter profile, and discuss the
implication of our work.
