Unified leapfrog CDI-FDTD methods with weakly conditional stability and hybrid implicit-explicit schemes

Abstract

Recently proposed leapfrog complying-divergence implicit-finite-difference time-domain (CDI-FDTD) method features many advantages over other unconditionally stable FDTD methods, such as unconditional stability, complying-divergence, and efficient leapfrog scheme with reduced right-hand-side (RHS) flops. This article investigates leapfrog CDI-FDTD schemes with weakly conditional stability. Two novel efficient schemes are proposed for problems with fine structures on one and two dimensions, respectively, namely, the hybrid implicit-explicit (HIE) and weakly conditionally stable (WCS) leapfrog CDI-FDTD. Their update equations and stability analysis are presented. Unification of the two schemes with weakly conditional stability and the unconditionally stable leapfrog CDI-FDTD is also provided. With this unified formulation, different leapfrog CDI-FDTD schemes can be flexibly selected according to the structural features of the problems. Numerical results show that although the proposed schemes are conditionally stable, the proposed schemes have the advantages of both less dispersion errors and higher efficiency over unconditionally stable leapfrog CDI-FDTD. Moreover, since the proposed schemes are formulated in the fundamental form and have complying-divergence, compared to conventional WCS-FDTD and HIE-FDTD, they not only are more efficient but also have less error, especially near the source region.