  A new general relativistic radiation magnetohydrodynamical code KORAL, is
described, which employs the M1 scheme to close the radiation moment equations.
The code has been successfully verified against a number of tests. Axisymmetric
simulations of super-critical magnetized accretion on a non-rotating black hole
(a=0.0) and a spinning black hole (a=0.9) are presented. The accretion rates in
the two models are \dot M = 100-200 \dot M_Edd. These first general
relativistic simulations of super-critical black hole accretion are potentially
relevant to tidal disruption events and hyper-accreting supermassive black
holes in the early universe. Both simulated models are optically and
geometrically thick, and have funnels through which energy escapes in the form
of relativistic gas, Poynting flux and radiative flux. The jet is significantly
more powerful in the a=0.9 run. The net energy outflow rate in the two runs
correspond to efficiencies of 5% (a=0) and 33% (a=0.9), as measured with
respect to the mass accretion rate at the black hole. These efficiencies agree
well with those measured in previous simulations of non-radiative geometrically
thick disks. Furthermore, in the a=0.9 run, the outflow power appears to
originate in the spinning black hole, suggesting that the associated physics is
again similar in non-radiative and super-critical accretion flows. While the
two simulations are efficient in terms of total energy outflow, both runs are
radiatively inefficient. Their luminosities are only \sim 1-10 L_Edd, which
corresponds to a radiative efficiency \sim 0.1%. Interestingly, most of the
radiative luminosity emerges through the funnels, which subtend a very small
solid angle. Therefore, measured in terms of a local radiative flux, the
emitted radiation is highly super-Eddington.
