An Optimized Unconditionally Stable Approach for the Solution of Discretized Maxwell’s Equations

Abstract

Explicit finite-difference time-domain (FDTD) methods are well-established grid-based computational approaches for the solution of the time-dependent Maxwell’s equations that govern electromagnetic problems. As they describe time-evolving phenomena, iterative procedures that calculate the necessary field values at successive time instants constitute fundamental procedures of the corresponding algorithms. In order for these time-marching processes to remain stable and provide reliable results, the size of the selected time step (i.e., the distance between successive time instants) should not exceed a known stability limit. Fulfillment of this condition may render the implementation of explicit FDTD approaches inefficient in certain types of simulations, where a large number of iterations are necessary to be executed. Appropriate unconditionally stable FDTD techniques have been developed throughout the years for modeling such classes of problems, which enable the selection of larger time steps without sacrificing stability. These methods are implicit, but they can perform efficiently as well, since they require fewer iterations for a given time period. In this work, we present an optimized version of a four-stage split-step unconditionally stable algorithm that is characterized by minimized errors at selected frequency bands, depending on the problem under investigation. Unlike conventional FDTD schemes, the spatial operators do not comply with the standard, Taylor series-based definitions but are designed in a special fashion, so that they minimize an error formula that represents the inherent space-time errors. In this way, the performance of the unconditionally stable algorithm is upgraded by simply modifying the values of some constant coefficients, thus producing a more efficient alternative without augmentation of the involved computational cost. The properties of the proposed algorithm are assessed theoretically as well as via numerical simulations, and the improved performance ensured by the new spatial operators is verified.