Design of Controllable Dispersion-Error Techniques

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 

For the discretization of spatial derivatives, the particular method uses a family of central difference operators, while a parametric expression with an extra degree of freedom for the temporal derivatives is employed [57], In two dimensions, these approximants have the general forms 

Ai 2 / AS V 2 where the inequality between the numerical and physical wavenumber has been enforced. The prior expressions can be viewed as polynomials of the unknown coefficients whose values minimize (2.110). Focusing on and bearing in mind the condition R2 0, one deduces that the solution of 3vj/BR= 0 provides a value of R that minimizes the temporal error. More specifically, this process yields 

To construct the optimized spatial operator, the complicated angular dependence of v2D must be somehow circumvented. Since an error reduction over all propagation angles is desired, the usage of the integrated error terms is proven to be fairly convenient. Therefore, it is denoted that 

The unknown coefficients can be computed via the minimization of Vj)D at a preselected frequency, which is guaranteed by the vanishing of the V D gradient or 

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