  We present a numerical method which evolves a two-temperature, magnetized,
radiative, accretion flow around a black hole, within the framework of general
relativistic radiation magnetohydrodynamics. As implemented in the code KORAL,
the gas consists of two sub-components -- ions and electrons -- which share the
same dynamics but experience independent, relativistically consistent,
thermodynamical evolution. The electrons and ions are heated independently
according to a standard prescription from the literature for
magnetohydrodynamical turbulent dissipation. Energy exchange between the
particle species via Coulomb collisions is included. In addition, electrons
gain and lose energy and momentum by absorbing and emitting synchrotron and
bremsstrahlung radiation, and through Compton scattering. All evolution
equations are handled within a fully covariant framework in the relativistic
fixed-metric spacetime of the black hole. Numerical results are presented for
five models of low luminosity black hole accretion. In the case of a model with
a mass accretion rate $\dot{M}\sim10^{-8} \dot M_{\rm Edd}$, we find that
radiation has a negligible effect on either the dynamics or the thermodynamics
of the accreting gas. In contrast, a model with a larger $\dot{M}\sim 4\times
10^{-4} \dot M_{\rm Edd}$ behaves very differently. The accreting gas is much
cooler and the flow is geometrically less thick, though it is not quite a thin
accretion disk.
