  Orbital advection is a significant bottleneck in disk simulations, and a
particularly tricky one when used in connection with magnetohydrodynamics. We
have developed an orbital advection algorithm suitable for the induction
equation with magnetic potential. The electromotive force is split into
advection and shear term, and we find that we do not need an advective gauge as
solving the orbital advection implicitly precludes the shear term from
canceling the advection term. We prove and demonstrate the third order in time
accuracy of the scheme. The algorithm is also suited for non-magnetic problems.
Benchmarked results of (hydrodynamical) planet-disk interaction and the of the
magnetorotational instability are reproduced. We include detailed descriptions
of the construction and selection of stabilizing dissipations (or high
frequency filters) needed to generate practical results. The scheme is
self-consistent, accurate, and elegant in its simplicity, making it
particularly efficiently in straightforward finite-difference methods. As a
result of the work, the algorithm is incorporated in the public version of the
Pencil Code, where it can be used by the community.
