The moving mesh method, as implemented in the <font size="2">AREPO</font> code, solves the ideal magnetohydrodynamic (MHD) equations on a Voronoi mesh using the finite volume method.
The mesh itself moves with the gas flow, which reduces advection errors and its quasi-Lagrangian nature offers a flexible spatial resolution.
In principle, the method should be particularly suitable for studying rotating disks as it can absorb the rotational motion in the mesh motion.
However, the motion and permanent distortion of the grid cells due to differential rotation introduced numerical noise in the original version of <font size="2">AREPO</font>, which has significantly reduced the usefulness of the code for cold disks.
By integrating the flux function more precisely over the interfaces of neighbouring cells, we show that the grid noise can be eliminated, greatly increasing the accuracy of the code for shear flows.
We apply the modified code to study the magnetorotational instability (MRI) and gravitational instability (GI) in the shearing box approximation and find good agreement with previous results from the literature, both in the linear and nonlinear regimes.
In global, two-dimensional simulations, the improved code resolves the Rossby wave instability (RWI) with similar accuracy as a static grid code on a polar mesh.
We conclude that the moving mesh method with the more accurate flux integration is well suited for the study of rotationally supported disks, particularly cold disks.
Future studies can benefit from the novel implementation of the shearing box approximation, which allows for adaptive spatial resolution during the simulation, while global simulations can benefit from the quasi-Lagrangian nature of the method.