Numerical Dispersion Analysis of the Complex Envelope ADI-FDTD Algorithm and Associated Numerical Artifacts Kyung-Young Jung and Fernando L. Teixeira ElectroScience Lab. and Dept. of Electrical and Computer Engineering, The Ohio State University, 1320 Kinnear Road, Columbus, OH 43212, USA {jung.166, teixeira.5}@osu.edu Abstract We investigate some numerical artifacts of complex envelope (CE) alternating-direction-implicit finite- difference time-domain (ADI-FDTD) method. The CE-ADI-FDTD is an unconditionally stable method suited for relatively narrowband problems in highly refined meshes. The analysis is based on the numerical dispersion relation and we focus in particular on the problem of spurious charges and anomalous-mode propagation (negative-group- velocity modes with positive phase velocities). In contrast to conventional ADI-FDTD, spurious charges associated with non-divergence free fields in CE-ADI-FDTD are inherently time-harmonic. As a result, they produce secondary radiation. Numerical examples are provided to illustrate the effects of such spurious charges on CE-ADI- FDTD simulation results. 1. Introduction The complex envelope (CE) alternating-direction-implicit finite-difference time-domain algorithm (ADI- FDTD) [1-5] is very attractive for electromagnetic scattering and propagation problems involving narrowband wave problems in highly refined grids. In CE-ADI-FDTD, the carrier frequency information is incorporated analytically by means of modified field update equations in terms of the slowly-varying complex field envelopes. Hence, the numerical dispersion error in CE-ADI-FDTD is determined by the bandwidth to carrier frequency ratio rather than by the highest frequency of excitation. From the analysis of the numerical (grid) dispersion in CE-ADI-FDTD, we examine in detail two undesired numerical artifacts, viz., spurious charges and anomalous-mode propagation. It is found out that the effects of spurious charges are more deleterious to CE-ADI-FDTD simulations compared to ADI- FDTD simulations. Because of space limitations, we just present the basic steps of the analysis here. A more detailed analysis and extra results will be presented in a journal manuscript to be submitted. 2. A Brief on the Dispersion Analysis In CE-ADI-FDTD, the field update components are represented as carrier-frequency-modulated CE field components: ] ~ Re[ J t c e ω ⋅ = U U , where c ω represents a carrier frequency, 1 J − = , and U and U ~ indicate the actual field components and the CE field components, respectively. Maxwell’s curl equations in terms of CE components can be expressed in a matrix form as ( ) 6 6 ] 0 [ ~ ] [ ] [ J ~ = + + ∂ ∂ U U R I t c ω (1) with ( ) T z y x z y x H H H E E E , , , , , ~ = U and ] [ R is a 6 6 × matrix associated with the curl operator. Here, 6 ] [ I and 6 ] 0 [ are the 6 6 × identity matrix and the 6 6 × null matrix, respectively. Splitting the operator ] [ R into two sub-operators ] [ ], [ B A and adding a perturbation term [6, 7], we obtain one full time-step of the CE-ADI-FDTD update on the grid point ( ) z k y j x i Δ Δ Δ , , and the time step n : ( ) 6 , , , , 1 6 , , , , 1 6 6 2 6 ] 0 [ 2 ~ ~ ] [ ] [ ] [ J ~ ~ ] [ ] [ 2 J ] [ ] [ 2 J 4 ] [ = + + + + − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + + k j i n k j i n c k j i n k j i n c c B A I t B I A I t I u u u u ω Δ ω ω Δ (2)