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 numerical 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 Käppeli (2019) presented a von Neumann stability analysis of globally constraint-preserving DG schemes for CED up to fourth order. That paper 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 working DGTD schemes for CED that go up to fifth 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 fourth and higher orders of accuracy, 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; though the schemes still retain their formal order of accuracy with WENO-type limiters. 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 second finding ensures that DGTD schemes retain optimal accuracy even in the presence of large spatial variations in permittivity and permeability. We also study the conservation of electromagnetic energy in these problems. 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. © 2019 Elsevier Inc.