Improvement Through the Dispersion Relation

This paragraph complements the analysis of Section 2.5 by presenting a technique for the construction of improved fourth-order spatial operators through the use of the discrete dispersion relation. Principally, the algorithm considers the ordinary leapfrog scheme for time marching, while it involves the parametric expression of (2.107) for spatial differentiation. By substituting plane-wave constituents in Maxwell s equations, the 2-D dispersion relation for an isotropic medium is 

The basic idea of the algorithm stems from the observation that the requirement knum = k is too strict to fulfill for all frequencies and angles of propagation. To circumvent this problem, an approximate version of (5.39), based on Taylor expansion, is utilized and then knum = k is applied in the modified equation [28], The difference between the two sides of the expression, so extracted, is defined as the error function 2D  

The next step is the treatment of the second-order quantities in (5.40). Nonetheless, the presence of ip entails the selection of certain propagation directions along which the extraction of the respective conditions would be conducted. Rather than proceeding to this restrictive approach, from an applicability viewpoint, a more general procedure for error reduction over all angles is developed. In this context, the remaining term 

the requirement v2D = 0 is equivalent to the nullification of the three harmonic terms in (5.43), where (5.41) has been partially utilized for the sake of clarity. To derive additional equations for the calculation of A and Bt, the first two terms of error indicator (5.43) are set to zero. In particular, if the constant term (i.e., the mean value of v2d) is zeroed, it is obtained that 

For the simplification of the previous expressions, it has been presumed that cell dimensions satisfy r = Ay/Ax, with the time-step fulfilling 

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