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 100-million-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 .
Because the internal thermal and gravitational energies of these massive , radiation-pressure-dominated fragments nearly cancel , small errors in energy conservation can cause spurious results such as spontaneous dissolution of isolated bodies , unless special care is taken .
This is likely to be a challenge for all eulerian codes in self-gravitating regimes where radiation pressure dominates .