  Neutrinos play a crucial role in the core-collapse supernova (CCSN) explosion
mechanism. The requirement of accurately calculating the transport of neutrinos
makes simulations of the CCSN mechanism extremely challenging and
computationally expensive. Historically, this stiff challenge has been met by
making approximations to the full transport equation. In this work, we compare
CCSN simulations in one- and two-dimensions with three approximate neutrino
transport schemes, each implemented in the {\tt FLASH} simulation framework. We
compare a two moment M1 scheme with an analytic closure (M1), the Isotropic
Diffusion Source Approximation (IDSA), and the Advanced Spectral Leakage (ASL)
method. We identify and discuss the advantages and disadvantages of each
scheme. For each approximate transport scheme, we use identical grid setups,
hydrodynamics, and gravity solvers to investigate the transport effects on
supernova shock dynamics and neutrino quantities. We find that the transport
scheme has a small effect on the evolution of protoneutron star (PNS) radius,
PNS mass, and the mass accretion rate. The neutrino luminosities, mean
energies, and shock radii have a $\sim$ 10-20\% quantitative difference but the
overall qualitative trends are fairly consistent between all three
approximations. We find larger differences in the gain region properties,
including the gain region mass and the net heating rate in the gain region, as
well as the strength of PNS convection in the core. We investigate the
progenitor, nuclear equation of state, and stochastic perturbation dependence
of our simulations and find similar magnitudes of impact on key quantities. We
also compare the computational expense of the various approximations.
