  Global, general relativistic magnetohydrodynamic (GRMHD) simulations of
nonradiative, magnetized disks are widely used to model accreting black holes.
We have performed a convergence study of GRMHD models computed with HARM3D. The
models span a factor of 4 in linear resolution, from 96x96x64 to 384x384x256.
We consider three diagnostics of convergence: (1) dimensionless shell-averaged
quantities such as plasma \beta; (2) the azimuthal correlation length of fluid
variables; and (3) synthetic spectra of the source including synchrotron
emission, absorption, and Compton scattering. Shell-averaged temperature is,
except for the lowest resolution run, nearly independent of resolution;
shell-averaged plasma \beta\ decreases steadily with resolution but shows signs
of convergence. The azimuthal correlation lengths of density, internal energy,
and temperature decrease steadily with resolution but show signs of
convergence. In contrast, the azimuthal correlation length of magnetic field
decreases nearly linearly with grid size. We argue by analogy with local
models, however, that convergence should be achieved with another factor of 2
in resolution. Synthetic spectra are, except for the lowest resolution run,
nearly independent of resolution. The convergence behavior is consistent with
that of higher physical resolution local model (shearing box) calculations and
with the recent nonrelativistic global convergence studies of Hawley et al.
(2011).
