Derivative Matching with Fictitious Points

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. 

Starting from the 1-D flux tensor form of Maxwell s equations 3Q/ 3/ = R 3Q/3 with Q = [Ez Hy]T, it is assumed that the interface between media A and B is located at i = E. As a consequence, the constitutive parameter matrix R becomes R = Ra for i L and R = Rr 

FIGURE 2.7 Positions of the real and fictitious lattice points in the vicinity of a material interface 

For the discretization of the 2M jump conditions, the algorithm uses central finite-difference schemes at each mesh node to construct 2M x 2M algebraic equations. For instance, let us consider jump condition 

Extension to the 2-D TM case opts for the vector form of 3Q/31= R 3Q/Bx + ZdQJdy with Q = HxHyEz T. If the material interface is again placed at i = L, constitutive parameter matrices R and Z receive the appropriate values at media A and B, as in the 1-D problem. Herein, derivative matching is performed for dEJdx and ()Hy/()x with the ensuing jump conditions across the interface 

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As can be deduced, for m > 2, expression (2.67) leads to cross derivatives by x and y, whose evaluation is rather cumbersome. To alleviate this difficulty, only one fictitious point can be considered at each side of the interface and hence only the zero- and first-order jump conditions are implemented. While this notion gives reliable solutions, an alternative quasi-fourth-order strategy has been presented in [28] for the consideration of higher order conditions and crossderivative computation. A fairly interesting feature of the derivative matching method is that it encompasses various schemes with different orders that permit its hybridization with other high-accuracy time-domain approaches. 

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