  (Abridged) We present one of the first physically-motivated two-dimensional
general relativistic magnetohydrodynamic (GRMHD) numerical simulations of a
radiatively-cooled black-hole accretion disk. The fiducial simulation combines
a total-energy-conserving formulation with a radiative cooling function, which
includes bremsstrahlung, synchrotron, and Compton effects. By comparison with
other simulations we show that in optically thin advection-dominated accretion
flows, radiative cooling can significantly affect the structure, without
necessarily leading to an optically thick, geometrically thin accretion disk.
We further compare the results of our radiatively-cooled simulation to the
predictions of a previously developed analytic model for such flows. For the
very low stress parameter and accretion rate found in our simulated disk, we
closely match a state called the "transition" solution between an outer
advection-dominated accretion flow and what would be a magnetically-dominated
accretion flow (MDAF) in the interior. The qualitative and quantitative
agreement between the numerical and analytic models is quite good, with only a
few well-understood exceptions. According to the analytic model then, at
significantly higher stress or accretion, we would expect a full MDAF to form.
The collection of simulations in this work also provide important data for
interpreting other numerical results in the literature, as they span the most
common treatments of thermodynamics, including simulations evolving: 1) the
internal energy only; 2) the internal energy plus an explicit cooling function;
3) the total energy without cooling; and 4) total energy including cooling. We
find that the total energy formulation is a necessary prerequisite for proper
treatment of radiative cooling in MRI accretion flows.
