As originally formulated, the Cartesian grids used in the FDTD method dictate that a smoothly varying surface must be approximated by one that is ``staircased.'' This approximation may lead to significant errors in certain problems [46,47]. Furthermore, if an object under consideration has small-scale structure, such as a narrow slot, the original method would have to use an excessively fine grid to accurately model the associated fields. To address these shortcomings, several solutions have been proposed.
If the object under consideration is more naturally described in an orthogonal coordinate system other than Cartesian, it is rather simple to develop update equations appropriate for that coordinate system as was done by Merewether in 1971 [11] and by Holland in 1983 [48]. Alternatively, a grid that uses varying spatial increments along the different coordinate directions can be used. In general, for a Cartesian grid, this results in rectangular cells and permits finer discretization in areas of rapid field fluctuation. Kunz and Lee [49,50] used this approach to calculate the external response on an aircraft to EMP. Monk and Süli have shown that this scheme preserves the second-order accuracy of the original algorithm [51,52]. Furthermore, subdomains can be gridded more finely than the rest of the problem space. This type of ``subgridding,'' where information is passed between the coarse and fine grids, was put forward by a number of researchers [53,54,55,56]. An alternative subgridding scheme was proposed by Kunz and Simpson [57]. Their formulation requires two runs. The first is done for a coarse grid that spans the entire computational domain, while the second is done for the finely-gridded subdomain and takes its boundary values from the stored values calculated during the coarse simulation.
Following the work of Yee [58], Umashankar et al. [59] and Taflove et al. [60] derived update equations that were suitable for modeling sub-cellular structures such as wires, narrow slots, and lapped joints in conducting screens. These equations were obtained from the integral form of Faraday's law rather than the differential form and resulted in modified equations only for cells where the sub-cellular structure was present. Several other researchers, including Holland and Simpson [61,62], Gilbert and Holland [63], Demarest [64], Turner and Bacon [65], Riley and Turner [66,67], Oates and Shin [68], and Wang [69,70], have developed techniques to handle sub-cellular structures.