  We present results from two-dimensional, general relativistic, viscous,
radiation hydrodynamic numerical simulations of Shakura-Sunyaev thin disks
accreting onto stellar mass Schwarzschild black holes. We consider cases on
both the gas- and radiation-pressure-dominated branches of the thermal
equilibrium curve, with mass accretion rates spanning the range from $\dot{M} =
0.01 L_\mathrm{Edd}/c^2$ to $10 L_\mathrm{Edd}/c^2$. The simulations directly
test the stability of this standard disk model on the different branches. We
find clear evidence of thermal instability for all radiation-pressure-dominated
disks, resulting universally in the vertical collapse of the disks, which in
some cases then settle onto the stable, gas-pressure-dominated branch. Although
these results are consistent with decades-old theoretical predictions, they
appear to be in conflict with available observational data from black hole
X-ray binaries. We also find evidence for a radiation-pressure-driven
instability that breaks the unstable disks up into alternating rings of high
and low surface density on a timescale comparable to the thermal collapse.
Since radiation is included self-consistently in the simulations, we are able
to calculate lightcurves and power density spectra (PDS). For the most part, we
measure radiative efficiencies (ratio of luminosity to mass accretion rate)
close to 6%, as expected for a non-rotating black hole. The PDS appear as
broken power laws, with a break typically around 100 Hz. There is no evidence
of significant excess power at any frequencies, i.e. no quasi-periodic
oscillations are observed.
