  In this work we extended an energy-integrated neutrino transport method to
facilitate efficient, yet precise, modeling of compact astrophysical objects.
We particularly focus on core-collapse supernovae. We implemented a gray
neutrino-transport framework from the literature into FLASH and performed a
detailed evaluation of its accuracy in core-collapse supernova simulations.
Based on comparisons with results from simulations using energy-dependent
neutrino transport, we incorporated several improvements to the original
scheme. Our analysis shows that our gray neutrino transport method successfully
reproduces key aspects from more complex energy-dependent transport across a
variety of progenitors and equations of state. We find both qualitative and
reasonable quantitative agreement with multi-group M1 transport simulations.
However, the gray scheme tends to slightly favor shock revival. In terms of
gravitational wave and neutrino signals, there is a good alignment with the
energy-dependent transport, although we find 15-30% discrepancies in the
average energy and luminosity of heavy-lepton neutrinos. Simulations using the
gray transport are around four times faster than those using energy-dependent
transport.
