Accuracy of the Yee FDTD Scheme for Normal Incidence of Plane Waves on Dielectric and Magnetic Interfaces

TL;DR

This paper provides a comprehensive analysis of the Yee FDTD scheme's accuracy for simulating plane waves at dielectric and magnetic interfaces, including error quantification and effects of grid placement.

Contribution

It offers a unified treatment of dielectric and magnetic media, deriving error estimates and insights into grid placement effects on FDTD interface accuracy.

Findings

Staggered grid spreads material discontinuity over a transition layer.

Error estimates depend on impedance contrast and Courant number.

Transition-layer model predicts deviation directions and magnitudes.

Abstract

This paper analyzes the accuracy of the standard Yee finite-difference time-domain (FDTD) scheme for simulating normal incidence of harmonic plane waves on planar interfaces between lossless, linear, homogeneous, isotropic media. Unlike prior analyses limited to dielectric interfaces, we provide a unified treatment encompassing both dielectric and magnetic media. We consider two common FDTD interface models based on different staggered-grid placements of material parameters. For each, we derive discrete analogs of the Fresnel reflection and transmission coefficients by formulating effective boundary conditions that emerge from the Yee update equations. A key insight is that the staggered grid implicitly spreads the material discontinuity over a transition layer of one spatial step, leading to systematic deviations from exact theory. We quantify these errors via a transition-layer model and provide (i) qualitative criteria predicting the direction and nature of deviations, and (ii) rigorous error estimates for both weak and strong impedance contrasts. Finally, we examine the role of the Courant number in modulating these errors, revealing conditions under which numerical dispersion and interface discretization jointly influence accuracy. This analysis provides quantitative error estimates that are directly applicable to simulation practice, offers a transition-layer interpretation that bridges classical FDTD with modern immersed-interface methods, and establishes benchmarks for validating structure-preserving discretizations of Maxwell's equations.