  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.
