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

A convergent second-order Cartesian grid finite difference scheme for the s olution of Maxwell's equations is presented. The scheme employs a staggered grid in space and represents the physical location of the material and met allic boundaries correctly, hence eliminating problems caused by staircasin g, and, contrary to the popular Yee scheme, enforces the correct jump-condi tions on the field components across material interfaces. A detailed analys is of the accuracy of the new embedding scheme is presented, confirming its second-order global accuracy. Furthermore, the scheme is proven to be a bo unded error scheme and thus convergent. Conditions for fully discrete stabi lity is furthermore established. This enables the derivation of bounds for fully discrete stability with CFL-restrictions being almost identical to th ose of the much simpler Yee scheme. The analysis exposes that the effects o f 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-dime nsional material interfaces, the Yee scheme may exhibit local divergence an d 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 ove r the Yee scheme, which in most cases exhibits First order accuracy. (C) 20 01 Academic Press.