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Grid-based magnetohydrodynamic (MHD) simulations have proven invaluable forthe study of astrophysical accretion disks. However, the fact that angularmomentum transport in disks is mediated by MHD turbulence (with structure downto very small scales) raises the concern that the properties of the modeledaccretion disks are affected by the finite numerical resolution of thesimulation. By implementing an orbital advection algorithm into the Athena codein cylindrical geometry, we have performed a set of global (but unstratified)Newtonian disk simulations extending up to resolutions previously unattained.We study the convergence of these models as a function of spatial resolutionand initial magnetic field geometry. The usual viscosity parameter ($\alpha$)or the ratio of thermal-to-magnetic pressure ($\beta$) are found to be poordiagnostics of convergence, whereas the average tilt angle of the magneticfield in the $(r,\phi)$-plane is a very good diagnostic of convergence. Wesuggest that this is related to the saturation of the MHD turbulence viaparasitic modes of the magnetorotational instability. Even in the case ofzero-net magnetic flux, we conclude that our highest resolution simulations(with 32-zones and 64-zones per vertical scale height) have achievedconvergence.  Our global simulations reach resolutions comparable to those used in local,shearing box models of MHD disk turbulence. We find that the saturationpredictors derived from local simulations correspond well to the instantaneouscorrelations between local flux and stress found in our global simulations.However, the conservation of magnetic flux implicit in local models is notrealized in our global disks. Thus, the magnetic connectivity of an accretiondisk represents physics that is truly global and cannot be captured in anyab-initio local model.

Global Simulations of Accretion Disks I: Convergence and Comparisons with Local Models