  We present spherically symmetric (1D) and axisymmetric (2D) supernova
simulations for a convection-dominated 9 Msun and a 20 Msun progenitor that
develops violent activity by the standing-accretion-shock instability (SASI).
We compare in detail the Aenus-Alcar code, which uses fully multidimensional
two-moment neutrino transport with an M1 closure, with a ray-by-ray-plus (RbR+)
version of this code and with the Prometheus-Vertex code that employs RbR+
two-moment transport with a Boltzmann closure. Besides testing consequences of
ignored non-radial neutrino-flux components in the RbR+ approximation, we also
discuss the influence of various transport ingredients applied or not applied
in recent literature, namely simplified neutrino-pair processes,
neutrino-electron scattering, velocity-dependent and gravitational-redshift
terms, and strangeness and many-body corrections for neutrino-nucleon
scattering. Alcar and Vertex show excellent agreement in 1D and 2D despite a
slightly but systematically smaller radius (~1km) and stronger convection of
the proto-neutron star with Alcar. As found previously, the RbR+ approximation
is conducive to explosions, but much less severely in the convection-dominated
9 Msun case than in the marginally exploding 20 Msun model, where the onset
time of explosion also exhibits big stochastic variations, and the RbR+
approximation has no distinctly stronger supportive effect than simplified pair
processes or strangeness and many-body corrections. Neglecting
neutrino-electron scattering has clearly unfavorable effects for explosions,
while ignoring velocity and gravitational-redshift effects can both promote or
delay the explosion. The ratio of advection timescale to neutrino-heating
timescale in 1D simulations is a sensitive indicator of the influence of
physics ingredients on explosions also in multidimensional simulations.
