Development of the Basic Conventional Algorithm

This section describes the main conventional higher order FDTD technique, presented in [3, 4, 8]. Retaining the usual notation, the members of the family are hereafter designated as (N, M), with the numbers in parentheses signifying the formal accuracy of temporal and spatial differentiation, respectively. For example, the simplest and most broadly implemented members 

As can be observed, the direct calculation of the 83 /8 /3 derivatives is rather laborious because they require extra time levels. To circumvent this difficulty, they are converted into spatial analogs, through repeated differentiation and consecutive use of Maxwell s equations, as 

It is important to state that the derivation of (2.17) requires the medium to be locally homogeneous i.e., it is assumed that all constitutive parameters are spatially independent. The small-valued (At)3/24 term in (2.15) and (2.16) permits further simplifications. Since all the derivatives encountered on the right-hand side of (2.17) are multiplied by it, they can be amply discretized by the common second-order finite differences. Therefore, one avoids ambiguous mathematical complexities and at the same time manages to construct a significantly improved FDTD approach, presenting lower phase and propagation errors. 

To attain the fourth-order spatial discretization of (2.15) and (2.16), the following central finite-difference schemes are employed. The former is the Yee s staggered-grid arrangement, while the latter is the collocated-mesh configuration, where component f and its derivatives are positioned at the same node. So, 

Coefficients Cs are determined via the Taylor expansion of the right-hand side of (2.20) and the requirement for minimum truncation error. Application of this process results in the M/2 x M/2 system 

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