  Self-consistent, time-dependent supernova (SN) simulations in three spatial
dimensions (3D) are conducted with the Aenus-Alcar code, comparing, for the
first time, calculations with fully multi-dimensional (FMD) neutrino transport
and the ray-by-ray-plus (RbR+) approximation, both based on a two-moment solver
with algebraic M1 closure. We find good agreement between 3D results with FMD
and RbR+ transport for both tested grid resolutions in the cases of a 20
solar-mass progenitor, which does not explode with the employed simplified set
of neutrino opacities, and of an exploding 9 solar-mass model. This is in stark
contrast to corresponding axisymmetric (2D) simulations, which confirm previous
claims that the RbR+ approximation can foster explosions in 2D in particular in
models with powerful axial sloshing of the stalled shock due to the standing
accretion shock instability (SASI). However, while local and instantaneous
variations of neutrino fluxes and heating rates can still be considerably
higher with RbR+ transport in 3D, the time-averaged quantities are very similar
to FMD results because of the absence of a fixed, artificial symmetry axis that
channels the flow. Therefore, except for stochastic fluctuations, the neutrino
signals and the post-bounce evolution of 3D simulations with FMD and RbR+
transport are also very similar, in particular for our calculations with the
better grid resolution. Higher spatial resolution has clearly a more important
impact than the differences by the two transport treatments. Our results back
up the use of the RbR+ approximation for neutrino transport in 3D SN modeling.
