Testing Convergence for Global Accretion Disks

Global disk simulations provide a powerful tool for investigating accretion and the underlying magnetohydrodynamic turbulence driven by magneto-rotational instability (MRI). Using them to accurately predict quantities such as stress, accretion rate, and surface brightness profile requires that purely numerical effects, arising from both resolution and algorithm, be understood and controlled. We use the flux-conservative Athena code to conduct a series of experiments on disks having a variety of magnetic topologies to determine what constitutes adequate resolution. We develop and apply several resolution metrics: 〈 Q z 〉 and 〈 Q ϕ 〉, the ratio of the grid zone size to the characteristic MRI wavelength, α mag , the ratio of the Maxwell stress to the magnetic pressure, and , the ratio of radial to toroidal magnetic field energy. For the initial conditions considered here, adequate resolution is characterized by 〈 Q z 〉 ⩾ 15, 〈 Q ϕ 〉 ⩾ 20, α mag ≈ 0.45, and . These values are associated with ⩾35 zones per scaleheight H , a result consistent with shearing box simulations. Numerical algorithm is also important. Use of the Harten–Lax–van Leer–Einfeldt flux solver or second-order interpolation can significantly degrade the effective resolution compared to the Harten–Lax–van Leer discontinuities flux solver and third-order interpolation. Resolution at this standard can be achieved only with large numbers of grid zones, arranged in a fashion that matches the symmetries of the problem and the scientific goals of the simulation. Without it, however, quantitative measures important to predictions of observables are subject to large systematic errors.