Fourth order compact implicit method for the Maxwell equations with discontinuous coefficients

We consider fourth order accurate compact schemes for numerical solutions to the Maxwell equations. The same mesh stencil is used as in the standard Yee scheme. In particular extra information over a wider stencil is not required. This has several advantages. First, it is relatively easy to modify an existing code based on the Yee algorithm to make it fourth order accurate. Second, a staggered mesh, without additional mesh locations, makes the boundary treatment easier since some of the quantities are located inside the domain rather than on the boundary. Also, a staggered grid system gives a lower error than a similar non-staggered system. The extension to dielectric materials is presented. This uses a compact implicit smoothing operator to redefine the piecewise constant dielectric coefficients.