  In this paper we present a full-fledged scheme for the second order accurate,
divergence-free evolution of vector fields on an adaptive mesh refinement (AMR)
hierarchy. We focus here on adaptive mesh MHD. The scheme is based on making a
significant advance in the divergence-free reconstruction of vector fields. In
that sense, it complements the earlier work of Balsara and Spicer (1999) where
we discussed the divergence-free time-update of vector fields which satisfy
Stoke's law type evolution equations. Our advance in divergence-free
reconstruction of vector fields is such that it reduces to the total variation
diminishing (TVD) property for one-dimensional evolution and yet goes beyond it
in multiple dimensions. Divergence-free restriction is also discussed. An
electric field correction strategy is presented for use on AMR meshes. The
electric field correction strategy helps preserve the divergence-free evolution
of the magnetic field even when the time steps are sub-cycled on refined
meshes. The above-mentioned innovations have been implemented in Balsara's
RIEMANN framework for parallel, self-adaptive computational astrophysics which
supports both non-relativistic and relativistic MHD. Several rigorous, three
dimensional AMR-MHD test problems with strong discontinuities have been run
with the RIEMANN framework showing that the strategy works very well.
