  Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations
in more than one space dimension must confront the challenge of controlling
errors in the discrete divergence of the magnetic field. One approach that has
been shown successful in stabilizing MHD calculations are constrained transport
(CT) schemes. CT schemes can be viewed as predictor-corrector methods for
updating the magnetic field, where a magnetic field value is first predicted by
a method that does not exactly preserve the divergence-free condition on the
magnetic field, followed by a correction step that aims to control these
divergence errors. In Helzel et al. (2011) the authors presented an unstaggered
constrained transport method for the MHD equations on 3D Cartesian grids. In
this work we generalize the method of Helzel et al. (2011) in three important
ways: (1) we remove the need for operator splitting by switching to an
appropriate method of lines discretization and coupling this with a
non-conservative finite volume method for the magnetic vector potential
equation, (2) we increase the spatial and temporal order of accuracy of the
entire method to third order, and (3) we develop the method so that it is
applicable on both Cartesian and logically rectangular mapped grids. The
evolution equation for the magnetic vector potential is solved using a
non-conservative finite volume method. The curl of the magnetic potential is
computed via a third-order accurate discrete operator that is derived from
appropriate application of the divergence theorem and subsequent numerical
quadrature on element faces. Special artificial resistivity limiters are used
to control unphysical oscillations in the magnetic potential and field
components across shocks. Test computations are shown that confirm third order
accuracy for smooth test problems and high-resolution for test problems with
shock waves.
