Analysis of the Divergence Properties for the Three-Dimensional Leapfrog ADI-FDTD Method

Abstract

Analysis of the divergence properties for the three-dimensional (3-D) leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is presented in this paper. The analysis of divergence properties is based on the Fourier domain plane wave method for dispersion analysis but extended with additional procedure to determine the homogeneous solutions. The singular value decomposition (SVD) is used to obtain the solutions of the homogeneous system in a reliable manner. There are two linearly independent null-modes in the homogeneous system corresponding to the repeated zero singular values. To provide more physical insights, proper linear combinations of the two null-modes are determined as TE and TM modes with respect to the direction of discrete numerical space wavevector. The divergence of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> fields is analyzed by examining the TM mode, while the divergence of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> fields is analyzed by examining the TE mode. Numerical results for the divergence properties of the leapfrog ADI-FDTD method are presented and compared with the conventional ADI-FDTD method, along with numerical experiments to illustrate the existence of divergent waves. The leapfrog ADI-FDTD method is not divergence-free in the source-free region, as is the case for the conventional ADI-FDTD method. Furthermore, both methods have different divergence properties in general.