  The magnetorotational instability (MRI) is an important process in
sufficiently ionized accretion disks, as it can create turbulence that acts as
an effective viscosity, mediating angular momentum transport. Due to its local
nature, it is often analyzed in the shearing box approximation with Eulerian
methods, which otherwise would suffer from large advection errors in global
disk simulations. In this work, we report on an extensive study that applies
the quasi-Lagrangian, moving-mesh code AREPO, combined with the Dedner cleaning
scheme to control deviations from $\nabla\cdot B=0$, to the problem of
magnetized flows in shearing boxes. We find that we can resolve the analytical
linear growth rate of the MRI with mean background magnetic field well. In the
zero net flux case, there is a threshold value for the strength of the
divergence cleaning above which the turbulence eventually dies out, and in
contrast to previous Eulerian simulations, the strength of the MRI does not
decrease with increasing resolution. In boxes with larger vertical aspect ratio
we find a mean-field dynamo, as well as an active shear current effect that can
sustain MRI turbulence for at least 200 orbits. In stratified simulations, we
obtain an active $\alpha\omega$ dynamo and the characteristic butterfly
diagram. Our results compare well with previous results obtained with static
grid codes such as ATHENA. We thus conclude that AREPO represents an attractive
approach for global disk simulations due to its quasi-Lagrangian nature, and
for shearing box simulations with large density variations due to its
continuously adaptive resolution.
