Error analyses of the two-equations two-unknowns method for low-frequency FDTD problems

Abstract

Numerical solutions of low-frequency electromagnetic problems are not a simple task, due to the inherent nature of their electrically small structures. The wavelength is very small relative to the size of the objects considered in these problems. Therefore, extremely fine meshes have to be used in the numerical solutions to perform accurate geometrical modeling of the scatterers and/or antennas. This requires extremely small unit time steps to preserve the stability of time-domain methods such as FDTD. As a result, a huge number of time iterations are required, leading to long and unacceptable computation times for today's computer technology. Although different modifications of the FDTD algorithms have been proposed to overcome this limitation, additional speed-up techniques are needed to further reduce the computation times. The two-equations two-unknowns (2E-2U) method is a good candidate for this purpose. In this study, error analyses of the 2E-2U method for low-frequency FDTD solutions are performed in detail, and an illustrative three-dimensional scattering problem of a dielectric sphere is solved for this purpose. Critical cases for the efficient application of the 2E-2U method are clearly revealed, and some important topics are highlighted.