Numerical Stability Analysis of M1-D ADI-FDTD Method for Coupled Transmission Lines

Abstract

This paper presents the numerical stability analysis of multiple one dimensional alternating direction implicit finite-difference time-domain (M1-D ADI-FDTD) method for coupled transmission lines. Two ways of splitting the operator matrices are attempted for the M1-D ADI-FDTD method. It is shown that the direct way of splitting based on self-mutual separation will reduce naturally to the equations for uncoupled transmission lines when the mutual terms are omitted. They also lead to update equations with left-hand sides involving tridiagonal matrices which can be solved efficiently. However, using the split matrices based on self-mutual separation, the resultant M1-D ADI-FDTD method for coupled transmission lines is not unconditionally stable. This is demonstrated through numerical stability analysis based on von Neumann method. Meanwhile, the proposed alternative way of splitting will also lead to update equations with left-hand sides involving tridiagonal matrices. Further numerical stability analysis is complemented to show that the proposed method is capable of treating coupled transmission lines using time step larger than the Courant-Friedrichs-Lewy (CFL) limit.