The magneto-rotational instability (MRI) is one of the most important processes in sufficiently ionized astrophysical disks. Grid-based simulations, especially those using the local shearing box approximation, provide a powerful tool to study the nonlinear turbulence the MRI produces. On the other hand, meshless methods have been widely used in cosmology, galactic dynamics, and planet formation, but have not been fully deployed on the MRI problem. We present local unstratified and vertically stratified MRI simulations with two meshless MHD schemes: a recent implementation of SPH MHD (Price 2012), and a MFM MHD scheme with constrained gradient divergence cleaning, as implemented in the GIZMO code (Hopkins 2017). Concerning variants of the SPH hydro force formulation, we consider both the "vanilla" SPH and the PSPH variant included in GIZMO. We find, as expected, that the numerical noise inherent in these schemes affects turbulence significantly. Also a high-order kernel, free of the pairing instability, is necessary. Both schemes adequately simulate MRI turbulence in unstratified shearing boxes with net vertical flux. The turbulence, however, dies out in zero-net-flux unstratified boxes, probably due to excessive numerical dissipation. In zero-netflux vertically stratified simulations, MFM can reproduce the MRI dynamo and its characteristic butterfly diagram for several tens of orbits before ultimately decaying. In contrast, extremely strong toroidal fields, as opposed to sustained turbulence, develop in equivalent simulations using SPH MHD. The latter unphysical state is likely caused by a combination of excessive artificial viscosity, numerical resistivity, and the relatively large residual errors in the divergence of the magnetic field.

In the early stages of a protoplanetary disk, when its mass is a significant fraction of its star's, turbulence generated by gravitational instability (GI) should feature significantly in the disk's evolution. At the same time, the disk may be sufficiently ionised for magnetic fields to play some role in the dynamics. Though usually neglected, the impact of magnetism on the GI may be critical, with consequences for several processes: the efficiency of accretion, spiral structure formation, fragmentation, and the dynamics of solids. In this paper, we report on global three-dimensional magnetohydrodynamical simulations of a self-gravitating protoplanetary disk using the meshless finite mass (MFM) Lagrangian technique. We confirm that GI spiral waves trigger a dynamo that amplifies an initial magnetic field to nearly thermal amplitudes (plasma β < 10), an order of magnitude greater than that generated by the magnetorotational instability alone. We also determine the dynamo's nonlinear back reaction on the gravitoturbulent flow: the saturated state is substantially hotter, with an associated larger Toomre parameter and weaker, more 'flocculent' spirals. But perhaps of greater import is the dynamo's boosting of accretion via a significant Maxwell stress; mass accretion is enhanced by factors of several relative to either pure GI or pure MRI. Our simulations use ideal MHD, an admittedly poor approximation in protoplanetary disks, and thus future studies should explore the full gamut of non-ideal MHD. In preparation for that, we exhibit a small number of Ohmic runs that reveal that the dynamo, if anything, is stronger in a non-ideal environment. This work confirms that magnetic fields are a potentially critical ingredient in gravitoturbulent young disks, possibly controlling their evolution, especially via their enhancement of (potentially episodic) accretion.

We carry out simulations of gravitationally unstable disks using smoothed particle hydrodynamics(SPH) and the novel Lagrangian meshless finite mass (MFM) scheme in the GIZMO code (Hopkins 2015). Our aim is to understand the cause of the non-convergence of the cooling boundary for fragmentation reported in the literature. We run SPH simulations with two different artificial viscosity implementations, and compare them with MFM, which does not employ any artificial viscosity. With MFM we demonstrate convergence of the critical cooling time scale for fragmentation at β crit ≈ 3.. Non-convergence persists in SPH codes, although it is significantly mitigated with schemes having reduced artificial viscosity such as inviscid SPH (ISPH) (Cullen & Dehnen 2010). We show how the non-convergence problem is caused by artificial fragmentation triggered by excessive dissipation of angular momentum in domains with large velocity derivatives. With increased resolution such domains become more prominent. Vorticity lags behind density due to numerical viscous dissipation in these regions, promoting collapse with longer cooling times. Such effect is shown to be dominant over the competing tendency of artificial viscosity to diminish with increasing resolution. When the initial conditions are first relaxed for several orbits, the flow is more regular, with lower shear and vorticity in non-axisymmetric regions, aiding convergence. Yet MFM is the only method that converges exactly. Our findings are of general interest as numerical dissipation via artificial viscosity or advection errors can also occur in grid-based codes. Indeed for the FARGO code values of β crit significantly higher than our converged estimate have been reported in the literature. Finally, we discuss implications for giant planet formation via disk instability.

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