Optimized three-dimensional FDTD discretizations of Maxwell’s equations on Cartesian grids

In this paper, novel finite-difference time-domain (FDTD) schemes are introduced for the numerical solution of Maxwell’s equations on dual staggered Cartesian three-dimensional lattices. The proposed techniques are designed to accomplish optimized performance according to certain features and requirements dictated by the investigated problems, thus making efficient use of the available computational resources. Starting from only few initial assumptions, a construction process based on the minimization of specific error formulae is developed, which is later exploited to derive the final form of the finite-difference operators. Previously, an elaborate analysis of the proposed indicators is provided, targeting at global error control over all propagation angles. Our methodology guarantees upgraded flexibility, as accuracy can be maximized within either narrow or wider frequency bands, without practically inducing significant computational overhead. Attractive qualities such as high convergence rates are now the natural consequence of the effective design process, rather than the minimization of the truncation errors of the difference expressions. In fact, the proposed FDTD approaches verify the possibility to attain improved levels of accuracy, without resorting to the traditional – Taylor based – forms of the individual operators. A theoretical analysis of the inherent dispersion artifacts reveals the full potential of the new algorithms, while numerical tests and comparisons unveil their unquestionable merits in practical applications.