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We provide a systematic description of the steps necessary -- and of thepotential pitfalls to be encountered -- when implementing a two-moment schemewithin an Implicit-Explicit (IMEX) scheme to include radiative-transfercontributions in numerical simulations of general-relativistic(magneto-)hydrodynamics. We make use of the M1 closure, which provides an exactsolution for the optically thin and thick limit, and an interpolation betweenthese limits. Special attention is paid to the efficient solution of theemerging set of implicit conservation equations. In particular, we present anefficient method for solving these equations via the inversion of a $4\times4$-matrix within an IMEX scheme. While this method relies on a fewapproximations, it offers a very good compromise between accuracy andcomputational efficiency. After a large number of tests in special relativity,we couple our new radiation code, \texttt{FRAC}, with the general-relativisticmagnetohydrodynamics code \texttt{BHAC} to investigate the radiative Michelsolution, namely, the problem of spherical accretion onto a black hole in thepresence of a radiative field. By performing the most extensive exploration ofthe parameter space for this problem, we find that the accretion's efficiencycan be expressed in terms of physical quantities such as temperature, $T$,luminosity, $L$, and black-hole mass, $M$, via the expression$\varepsilon=(L/L_{\rm Edd})/(\dot{M}/\dot{M}_{\rm Edd})= 7.41\times10^{-7}\left(T/10^6\,\mathrm{K}\right)^{0.22} \left(L/L_\odot\right)^{0.48}\left(M/M_\odot\right)^{0.48}$, where $L_{\mathrm{Edd}}$ and$\dot{M}_{\mathrm{Edd}}$ are the Eddington luminosity and accretion rate,respectively. Finally, we also consider the accretion problem away fromspherical symmetry, finding that the solution is stable under perturbations inthe radiation field.

Two-moment scheme for general-relativistic radiation hydrodynamics: a systematic description and new applications