Development of the Three-Dimensional Algorithm

Let us express Maxwell s hyperbolic time-dependent laws in vector notation  

The above dispersion-optimized single-directional design process repeated along every mesh axis is proven fairly rigorous and convergent, even for coarse lattice resolutions. Hence, spatial derivatives receive an enhanced manipulation without augmenting overall complexity, since (6.42)—(6.46) are computed only once, ahead of the primary simulation phase. 

Concerning the explicit update of Fa(u t) in (6.36), the following modified leapfrog scheme is utilized 

The stability of (6.48) and (6.49) is mathematically examined by the von Neumann and the energy inequalities method [38]. According to the latter, the inner product on both sides of (6.36) with F +1 + F gives 

FIGURE 6.5 Evolution of the E component for different FDTD and WENO-TD configurations 

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