  In this paper we study the problem of divergence-free numerical MHD and show
that the work done so far still has four key unresolved issues. We resolve
those issues in this paper. The problem of reconstructing MHD flow variables
with spatially second order accuracy is also studied. The other goal of this
paper is to show that the same well-designed second order accurate schemes can
be formulated for more complex geometries such as cylindrical and spherical
geometry. Being able to do divergence-free reconstruction in those geometries
also resolves the problem of doing AMR in those geometries. The resulting MHD
scheme has been implemented in Balsara's RIEMANN framework for parallel,
self-adaptive computational astrophysics. The present work also shows that
divergence-free reconstruction and the divergence-free time-update can be done
for numerical MHD on unstructured meshes. All the schemes designed here are
shown to be second order accurate. Several stringent test problems are
presented to show that the methods work, including problems involving high
velocity flows in low plasma-b magnetospheric environments.
