  We present a new, computationally efficient, energy-integrated approximation
for neutrino effects in hot and dense astrophysical environments such as
supernova cores and compact binary mergers and their remnants. Our new method,
termed ILEAS for Improved Leakage-Equilibration-Absorption Scheme, improves the
lepton-number and energy losses of traditional leakage descriptions by a novel
prescription of the diffusion time-scale based on a detailed energy integral of
the flux-limited diffusion equation. The leakage module is supplemented by a
neutrino-equilibration treatment that ensures the proper evolution of the total
lepton number and medium plus neutrino energies as well as neutrino-pressure
effects in the neutrino-trapping domain. Moreover, we employ a simple and
straightforwardly applicable ray-tracing algorithm for including re-absorption
of escaping neutrinos especially in the decoupling layer and during the
transition to semi-transparent conditions. ILEAS is implemented on a
three-dimensional (3D) Cartesian grid with a minimum of free and potentially
case-dependent parameters and exploits the basic physics constraints that
should be fulfilled in the neutrino-opaque and free-streaming limits. We
discuss a suite of tests for stationary and time-dependent proto-neutron star
models and post-merger black-hole-torus configurations, for which 3D ILEAS
results are demonstrated to agree with energy-dependent 1D and 2D two-moment
(M1) neutrino transport on the level of 10--15 percent in basic neutrino
properties. This also holds for the radial profiles of the neutrino
luminosities and of the electron fraction. Even neutrino absorption maps around
torus-like neutrino sources are qualitatively similar without any fine-tuning,
confirming that ILEAS can satisfactorily reproduce local losses and
re-absorption of neutrinos as found in sophisticated transport calculations.
