  Using a version of the ZEUS code, we carry out two-dimensional simulations of
self-gravitating shearing sheets, with application to QSO accretion disks at a
few thousand Schwarzschild radii, corresponding to a few hundredths of a parsec
for a 10^8 solar-mass black hole. Radiation pressure and optically thick
radiative cooling are implemented via vertical averages. We determine
dimensionless versions of the maximum surface density, accretion rate, and
effective viscosity that can be sustained by density-wave turbulence without
fragmentation. Where fragments do form, we study the final masses that result.
The maximum Shakura-Sunyaev viscosity parameter is approximately 0.4.
Fragmentation occurs when the cooling time is less than about twice the
shearing time, as found by Gammie and others, but can also occur at very long
cooling times in sheets that are strongly radiation-pressure dominated. For
accretion at the Eddington rate onto a 10^8 solar-mass black hole,
fragmentation occurs beyond four thousand Schwarzschild radii, r_s. Near this
radius, initial fragment masses are several hundred suns, consistent with
estimates from linear stability; final masses after merging increase with the
size of the sheet, reaching several thousand suns in our largest simulations.
With increasing black-hole mass at a fixed Eddington ratio, self-gravity
prevails to smaller multiples of r_s, where radiation pressure is more
important and the cooling time is longer compared to the dynamical time;
nevertheless, fragmentation can occur and produces larger initial fragment
masses. We also find energy conservation is likely to be a challenge for all
eulerian codes in self-gravitating regimes where radiation pressure dominates.
