Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution - Part I, second-order FVTD schemes

Overall, in this paper, we report a new scheme for numerically solving Maxwell's equations in material media, with special attention paid to a second-order-accurate formulation. Several numerical examples are presented to show that the proposed technique works. Because of its sub-cell resolving ability, the new method retains second-order accuracy even when material permeability and permittivity vary by an order-of-magnitude over just one or two zones. Furthermore, because the new method is also unconditionally stable in the presence of stiff source terms (i.e., in problems involving giant conductivity variations), it can handle several orders-of-magnitude variation in material conductivity over just one or two zones without any reduction of the time-step. Consequently, the CFL depends only on the propagation speed of light in the medium being studied.