Convergent Cartesian grid methods for Maxwell's equations in complex geometries

A convergent second-order Cartesian grid finite difference scheme for the solution of Maxwell's equations is presented. The scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly, hence eliminating problems caused by staircasing, and, contrary to the popular Yee scheme, enforces the correct jump-conditions on the field components across material interfaces.

A detailed analysis of the accuracy of the new embedding scheme is presented, confirming its second-order global accuracy. Furthermore, the scheme is proven to be a bounded error scheme and thus convergent. Conditions for fully discrete stability is furthermore established. This enables the derivation of bounds for fully discrete stability with CFL-restrictions being almost identical to those of the much simpler Yee scheme.

The analysis exposes that the effects of staircasing as well as a lack of properly enforced jump-conditions on the field components have significant consequences for the global accuracy. It is, among other things, shown that for cases in which a field component is discontinuous along a grid line, as happens at general two- and three-dimensional material interfaces, the Yee scheme may exhibit local divergence and loss of global convergence.

To validate the analysis several one- and two-dimensional test cases are presented, showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the Yee scheme, which in most cases exhibits first order accuracy.