  It is now generally agreed that multidimensional, multigroup, radiation
hydrodynamics is an indispensable element of any realistic model of
stellar-core collapse, core-collapse supernovae, and protoneutron star
instabilities. We have developed a new, two-dimensional, multigroup algorithm
that can model neutrino-radiation-hydrodynamic flows in core-collapse
supernovae. Our algorithm uses an approach that is similar to the ZEUS family
of algorithms, originally developed by Stone and Norman. However, we extend
that previous work in three significant ways: First, we incorporate
multispecies, multigroup, radiation hydrodynamics in a flux-limited-diffusion
approximation. Our approach is capable of modeling pair-coupled
neutrino-radiation hydrodynamics, and includes effects of Pauli blocking in the
collision integrals. Blocking gives rise to nonlinearities in the discretized
radiation-transport equations, which we evolve implicitly in time. We employ
parallelized Newton-Krylov methods to obtain a solution of these nonlinear,
implicit equations. Our second major extension to the ZEUS algorithm is
inclusion of an electron conservation equation, which describes evolution of
electron-number density in the hydrodynamic flow. This permits following the
effects of deleptonization in a stellar core. In our third extension, we have
modified the hydrodynamics algorithm to accommodate realistic, complex
equations of state, including those having non-convex behavior. In this paper,
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