Dispersion-Optimized Higher Order FDTD Techniques

5 DISPERSION-OPTIMIZED HIGHER ORDER FDTD TECHNIQUES 

The potential to exploit the advanced features of higher order FDTD schemes in the construction of algorithms, with reduced dispersion error at specific parts of the frequency spectrum, is indeed a challenging task. This section presents several robust methodologies that attain this critical issue from different perspectives. 

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This section deals with the construction of optimal higher order FDTD schemes with adjustable dispersion error. Rather than implementing the ordinary approaches, based on Taylor series expansion, the modified finite-difference operators are designed via alternative procedures that enhance the wideband capabilities of the resulting numerical techniques. First, an algorithm founded on the separate optimization of spatial and temporal derivatives is developed. Additionally, a second method is derived that reliably reflects artificial lattice inaccuracies via the necessary algebraic expressions. Utilizing the same kind of differential operators as the typical fourth-order scheme, both approaches retain their reasonable computational complexity and memory requirements. Furthermore, analysis substantiates that important error compensation... 

The most important feature of this dispersion-optimized FDTD method is the higher order nonstandard finite-difference schemes [6, 7] that substitute their conventional counterparts in the differentiation of Ampere s and Faraday s laws, as already described in (3.31). The proposed technique can be occasionally even 7 to 8 orders of magnitude more accurate than the fourth-order implementations of Chapter 2. Although the cost is slightly increased, the overall simulation benefits from the low resolutions and the reduced number of iterations. Thus, for spatial derivative approximation, the following two operators are defined ... 

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