  In numerical models of thin astrophysical disks that use an Eulerian scheme,
gas orbits supersonically through a fixed grid. As a result the time step is
sharply limited by the Courant condition. Also, because the mean flow speed
with respect to the grid varies with position, the truncation error varies
systematically with position. For hydrodynamic (unmagnetized) disks an
algorithm called FARGO has been developed that advects the gas along its mean
orbit using a separate interpolation substep. This relaxes the constraint
imposed by the Courant condition, which now depends only on the peculiar
velocity of the gas, and results in a truncation error that is more nearly
independent of position. This paper describes a FARGO-like algorithm suitable
for evolving magnetized disks. Our method is second order accurate on a smooth
flow and preserves the divergence-free constraint to machine precision. The
main restriction is that the magnetic field must be discretized on a staggered
mesh. We give a detailed description of an implementation of the code and
demonstrate that it produces the expected results on linear and nonlinear
problems. We also point out how the scheme might be generalized to make the
integration of other supersonic/super-fast flows more efficient. Although our
scheme reduces the variation of truncation error with position, it does not
eliminate it. We show that the residual position dependence leads to
characteristic radial variations in the density over long integrations.
