Development of the Higher Order Nonstandard Forms in Cartesian Coordinates

3 DEVELOPMENT OF THE HIGHER ORDER NONSTANDARD FORMS IN CARTESIAN COORDINATES 

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  

Similar operators with respect to y and z can be simply obtained. Correction function (k, A) accepts multiple arguments and therefore it can drastically contribute to the minimization of the undesirable FDTD dispersion and dissipation errors. Among its various possible forms, some of the most convenient are 

FIGURE 3.1 Graphical representation of difference operators. The numbers at the vertices indicate the sign of summation for the corresponding values of the approximated function f. The origin (x, y, z) is located at the center of the cell 

Parameters qi, for i = 1, 2, 3, are given by (3.40) and (3.41) in terms of condition (3.28). Note that despite their different notation, they have exactly the same function. In fact, their values certify the stable profile of the algorithm as well as its enhanced levels of convergence. 

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