  Computational electrodynamics (CED), the numerical solution of Maxwell's
equations, plays an incredibly important role in several problems in science
and engineering. High accuracy solutions are desired, and the discontinuous
Galerkin (DG) method is one of the better ways of delivering high accuracy in
CED. Maxwell's equations have a pair of involution constraints for which
mimetic schemes that globally satisfy the constraints at a discrete level are
highly desirable. Balsara and Kappeli presented a von Neumann stability
analysis of globally constraint-preserving DG schemes for CED up to 4'th order
which was focused on developing the theory and documenting the superior
dissipation and dispersion of DGTD schemes in media with constant permittivity
and permeability. In this paper we present DGTD schemes for CED that go up to
5'th order of accuracy and analyze their performance when permittivity and
permeability vary strongly in space. Our DGTD schemes achieve constraint
preservation by collocating the electric displacement and magnetic induction as
well as their higher order modes in the faces of the mesh. Our first finding is
that at 4'th and higher orders, one has to evolve some zone-centered modes in
addition to the face-centered modes. It is well-known that the limiting step in
DG schemes causes a reduction of the optimal accuracy of the scheme. In this
paper we document simulations where permittivity and permeability vary by
almost an order of magnitude without requiring any limiting of the DG scheme.
This very favorable finding ensures that DGTD schemes retain optimal accuracy
even in the presence of large spatial variations in permittivity/permeability.
Our third finding shows that the electromagnetic energy is conserved very well
even when permittivity and permeability vary strongly in space; as long as the
conductivity is zero.

Numerical Analysis;Computational Physics