Discretization of Maxwells Equations

When the time-dependent Maxwell s equations are to be discretized, the nature of vector difference operator requires the decomposition of the prospective L2[.] to guarantee the pertinent consistency. Hence, 

For the 3-D case, inspection of (3.28) reveals that the weighted superposition of d [.] for e (x, y, z) requires three independent difference operators, i.e., d [.] - provided, again, by (3.1) - as well as the two auxiliary ones d [.] and dj [.]. Hence, studying a certain partial derivative for illustration 3Z, the unknown operators d r [ ] and d [.] receive the form of 

In this manner, difference operators d [.] and [.] are fully determined, and so allowing the consistent design of (3.30) and the subsequent time update of the 3-D Maxwell s equations (3.31). Conclusively, a noteworthy feature of these operators is the use of extra nodal points for the approximation of partial derivatives. This implies that, unlike the limited stencil of the FDTD technique, the nonstandard concepts offer an enhanced manipulation of the elementary cells and through additional degrees of freedom permit the significant suppression of dispersion and anisotropy errors. These merits are much more prominent in higher order formulations, where the abruptly curved waveguide or antenna components, the arbitrary material discontinuities, and the dissimilar interfaces stipulate very robust simulations. 

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