  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 ensuing nonlinear turbulence. On the other hand, while
meshless methods have been widely used in both cosmology, galactic dynamics,
and planet formation they 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 (Price2012), and a MFM
MHD scheme with a constrained gradient divergence cleaning scheme, as
implemented in the GIZMO code \citep{Hopkins2017}. 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. A high order
kernel, free of the pairing instability, is necessary. Both schemes can
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 and numerical dissipation. In zero-net-flux
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. This
unphysical state in SPH MHD 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 remaining even after cleaning
procedures are applied.
