Higher Order FDTD Differentiation

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... 

The use offictitious points as a means of locally modifying the differential stencils near laborious media interfaces in finite-difference simulations has been initially developed in [17, 18] and extended in [21, 28]. The specific method, which matches the problematic boundaries with physical derivative conditions, enhances the flexibility of higher order FDTD schemes and facilitates the discretization of difficult geometries. 

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 ... 

Let us consider Lindman annihilators [3], which are constmcted through the use of projection operators incorporating past data at the boundary. Primarily, they involve the suitable field approximations by solving a system of partial differential equations in terms of certain correction functions. Focusing on the absorption of Ex electric component at the outer boundary, z = LAz, the higher order nonstandard FDTD form of its update expression for a lossy medium, in conjunction with (3.70) and (3.71), becomes... 

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