Accuracy Limitations of the Locally One-Dimensional FDTD Technique

Abstract

While the alternating-direction implicit finitedifference time-domain (ADI-FDTD) method preserves the second-order temporal accuracy of the conventional FDTD technique, the locally one-dimensional (LOD)-FDTD method exhibits a first-order in time splitting error. Despite this difference, the numerical dispersion analyses of these methods reveal that both present similar accuracy properties. For this reason, the characteristic non-commutativity error of the LOD-FDTD scheme has not received much attention. In this work, we determine the closed form of the local truncation error for the 3D-LOD-FDTD scheme. We find that it presents error terms which depend on the time-step size multiplied by the spatial derivatives of the fields. Numerical results confirm that these terms become a significant source of error which is not revealed in the dispersion analyses.

On the Behavior of the LOD-FDTD Method at Dielectric Interfaces

“…This phenomenon can be ascribed to the fact that the local truncation error of the LODFDTD method exhibits firstorder error terms that depend on and the spatial derivatives of the fields. These terms are not present in the ADI scheme [8]. Consequently, the conventional LODFDTD method should be avoided when one is concerned with the accurate computation of nearfields and/or the interpretation of physical phenomena involving them.…”

“…The accuracy of the solution in this type of problems is essentially dependent on the numerical dispersion error. Hence, other error sources tend to remain hidden [7], [8].…”

Second-Order Temporal-Accurate Scheme for 3-D LOD-FDTD Method With Three Split Matrices

“…Thus, the limitation related to first order temporal accuracy in [16] would be eliminated. Note that in (8), more noncommutative terms AC, CA, BC, CB are involved compared to only AB, BA in [4, (26)].…”

A New Hybrid Implicit–Explicit FDTD Method for Local Subgridding in Multiscale 2-D TE Scattering Problems

“…Examples of these implicit methods are Crank-Nicolson (CN) FDTD [9], alternating direction implicit (ADI) FDTD [10], locally one-dimensional (LOD) FDTD [11,12] and Laguerre-FDTD [13,14]. A lot of comparisons concerning the CPU-time and accuracy of these implicit schemes have been reported [15]- [17]. Compared to explicit methods, they all reduce the number of time steps per simulation at the expense of a higher computational effort per time step.…”