Three-dimensional, global, radiative GRMHD simulations of a thermally   unstable disc
  We present results of a set of three-dimensional, general relativistic radiation magnetohydrodynamics simulations of thin accretion discs around a non-rotating black hole to test their thermal stability. We consider two cases, one that is initially radiation pressure dominated and expected to be thermally unstable and another that is initially gas-pressure dominated and expected to remain stable. Indeed, we find that cooling dominates over heating in the radiation pressure dominated model, causing the disc to collapse vertically on roughly the local cooling timescale. We also find that heating and cooling within the disc have a different dependence on the mid-plane pressure, a prerequisite of thermal instability. Comparison of our data with the relevant thin-disc thermal equilibrium curve suggests that our disc may be headed for the thermally stable, gas-pressure-dominated branch. However, because the disc collapses to the point that we are no longer able to resolve it, we had to terminate the simulation. On the other hand, the gas pressure dominated model, which was run for twice as long as the radiation pressure dominated one, remains stable, with heating and cooling roughly in balance. Finally, the radiation pressure dominated simulation shows some evidence of viscous instability. The strongest evidence is in plots of surface density, which show the disc breaking up into rings. 
B.Mishra, P. C. Fragile, L. C. Johnson, W. Klu\'zniak
astro-ph.HE