Assessing Quantitative Results in Accretion Simulations: From Local to   Global
  Discretized numerical simulations are a powerful tool for investigation of nonlinear MHD turbulence in accretion disks. However, confidence in their quantitative predictions requires a demonstration that further refinement of the spatial gridscale would not result in any significant change. This has yet to be accomplished, particularly for global disk simulations. In this paper, we combine data from previously published stratified shearing box simulations and new global disk simulations to calibrate several quantitative diagnostics by which one can estimate progress toward numerical convergence. Using these diagnostics, we find that the established criterion for an adequate numerical description of linear growth of the magneto-rotational instability (the number of cells across a wavelength of the fastest-growing vertical wavenumber mode) can be extended to a criterion for adequate description of nonlinear MHD disk turbulence, but the standard required is more stringent. We also find that azimuthal resolution, which has not often been extensively examined in previous studies, can significantly affect the evolution of the poloidal magnetic field. We further analyze the comparative resolution requirements of a small sample of initial magnetic field geometries; not surprisingly, more complicated initial field geometries require higher spatial resolution. Otherwise, they tend to evolve to qualitatively similar states if evolved for sufficient time. Applying our quantitative resolution criteria to a sample of previously published global simulations, we find that, with perhaps a single exception, they are significantly under-resolved, and therefore underestimate the magnetic turbulence and resulting stress levels throughout the accretion flow. 
John F. Hawley, Xiaoyue Guan, and Julian H. Krolik
astro-ph.HE