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<!PCT!>
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.