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arxiv: 2512.14863 · v3 · submitted 2025-12-16 · 🧮 math.NA · cs.NA· math-ph· math.MP· physics.comp-ph

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

Pavel A. Makarov (1) , Vladimir I. Shcheglov (2) ((1) Institute of Physics , Mathematics , Komi Science Centre of the Ural Branch of the Russian Academy of Sciences , (2) Laboratory of magnetic phenomena in microelectronics , Kotelnikov Institute of Radioengineering , Electronics of Russian Academy of Sciences) This is my paper

Pith reviewed 2026-05-16 21:32 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath-phmath.MPphysics.comp-ph MSC 65M06
keywords Yee FDTDinterface accuracyFresnel coefficientstransition layernormal incidencenumerical dispersiondielectric interfacemagnetic interface
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The pith

The Yee FDTD scheme implicitly spreads material interfaces over one grid step, producing systematic deviations from exact Fresnel reflection and transmission.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the standard Yee finite-difference time-domain scheme for electromagnetic waves at planar interfaces between lossless media creates an effective transition layer one spatial step thick because of the staggered placement of electric and magnetic field components and material parameters. This layer replaces the intended sharp jump with averaged properties, so the discrete reflection and transmission coefficients differ from the exact continuous Fresnel formulas. The analysis covers both dielectric and magnetic interfaces through two common grid placements, derives the effective boundary conditions directly from the update equations, and supplies error bounds that grow with impedance contrast. A reader would care because FDTD simulations are used to predict wave behavior at boundaries; knowing the size and direction of these interface errors lets users judge when a grid is fine enough without running exhaustive tests.

Core claim

The staggered grid in the Yee scheme spreads the material discontinuity over a transition layer of one spatial step. This produces effective boundary conditions whose reflection and transmission coefficients deviate from the exact Fresnel values in a manner that can be predicted from the layer's averaged permittivity and permeability. The deviations are quantified for both weak and strong impedance contrasts, with the Courant number modulating the error through its interaction with numerical dispersion.

What carries the argument

The transition-layer model that converts the sharp interface into an effective one-cell-thick slab whose averaged material parameters determine the discrete Fresnel coefficients.

If this is right

  • For small impedance contrasts the direction of the reflection error is fixed by whether the material jump is placed on electric or magnetic nodes.
  • Strong contrasts produce larger relative errors that scale with the contrast ratio and require proportionally finer grids to control.
  • Raising the Courant number can either increase or decrease the interface error depending on whether numerical dispersion reinforces or opposes the layer effect.
  • The derived discrete coefficients supply concrete benchmarks for testing improved interface treatments in other Maxwell discretizations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same one-cell spreading effect is likely to appear in staggered-grid schemes for acoustic or elastic waves, suggesting a common source of interface error across wave equations.
  • Correcting the effective layer properties inside existing FDTD codes could reduce the observed deviations without changing the grid spacing.
  • In simulations containing many closely spaced interfaces the local errors may accumulate and shift the overall transmission spectrum by an amount larger than single-interface estimates predict.

Load-bearing premise

The analysis assumes that the two common staggered-grid placements of material parameters represent typical simulation setups for normal incidence.

What would settle it

Run a Yee FDTD simulation of a harmonic plane wave at a dielectric interface with known exact reflection coefficient, extract the numerical coefficient from the steady-state fields, and check whether the difference matches the transition-layer error estimate to within a few percent.

Figures

Figures reproduced from arXiv: 2512.14863 by (2) Laboratory of magnetic phenomena in microelectronics, Electronics of Russian Academy of Sciences), Komi Science Centre of the Ural Branch of the Russian Academy of Sciences, Kotelnikov Institute of Radioengineering, Mathematics, Pavel A. Makarov (1), Vladimir I. Shcheglov (2) ((1) Institute of Physics.

Figure 1
Figure 1. Figure 1: Orientation of electromagnetic field components in incident, reflected and transmitted waves [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Yee grid in the Yee grid is 2M × 2T, which are not completely independent of each other, since half of them are associated with the electric component of the field, and half—with the magnetic part. Thus, the total spatial extent of the considered region of space D is determined precisely by the number M, and the duration of the simulation is fixed by the number T. Let us now refine the approximation (14), … view at source ↗
Figure 3
Figure 3. Figure 3: FDTD-model of a planar interface between two dielectrics [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FDTD-model of a planar interface between two magnetics [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FDTD Fresnel coefficients for reflection [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Formation features of a transition layer between two dielectrics: [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FDTD-coefficients for reflection Re (curves 1) and transmission Te (curves 2 and 3) in the case of the interface between two media with slightly different impedances: η1/η2 = p 4/3 ≈ 1.16 (curves 2 for dielectrics and 3—for magnetics) and η1/η2 = p 3/4 ≈ 0.87 (curves 3 for dielectrics and 2—for magnetics) is clearly evident in the region of very poor discretization (when Nλ is sufficiently small), i.e., fo… view at source ↗
Figure 8
Figure 8. Figure 8: Influence on the errors of FDTD reflection coefficients (red curves) and transmission coefficients (blue lines) [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The difference between the errors of the FDTD coefficients for reflection [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: a) FDTD coefficients for reflection Re (curves 1) and transmission Te (curves 2 and 3) in the case of the interface between two media with a high difference in impedances: η1/η2 = 10 (curves 2 for dielectrics and 3 for magnetics) and η1/η2 = 0.1 (curves 3 for dielectrics and 2 for magnetics); b) Relative errors of the corresponding coefficients under the same simulating conditions poor simulation result. … view at source ↗
Figure 11
Figure 11. Figure 11: FDTD-model of “real interfaces” involve simultaneous changes in both [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper analyzes the accuracy of the standard Yee FDTD scheme for normal incidence of harmonic plane waves on planar dielectric and magnetic interfaces. It derives discrete analogs of the Fresnel reflection and transmission coefficients from the Yee update equations for two common staggered-grid placements of material parameters, interprets the discretization as implicitly creating a one-cell transition layer, quantifies systematic deviations from exact theory via this model, supplies qualitative criteria and rigorous error estimates for weak and strong impedance contrasts, and examines how the Courant number modulates the combined effects of numerical dispersion and interface discretization.

Significance. If the derivations are correct, the work supplies quantitative, directly usable error bounds for FDTD interface simulations that extend prior dielectric-only studies to magnetic media. The transition-layer interpretation offers a concrete bridge between classical Yee schemes and immersed-interface methods, while the Courant-number analysis identifies regimes where interface errors dominate or interact with dispersion. These results are of immediate value for validation benchmarks and for choosing simulation parameters in lossless media.

major comments (2)
  1. [§3] §3 (Derivation of discrete Fresnel coefficients): The effective boundary conditions are obtained by treating material parameters as pointwise assigned to staggered locations without averaging. This produces the claimed one-cell transition layer, but the manuscript does not demonstrate that the resulting error estimates remain valid (or provide conservative bounds) when common arithmetic or harmonic averaging is applied across the interface cell, which alters the effective impedance jump seen by the fields.
  2. [§5] §5 (Courant-number dependence): The analysis of how the Courant number modulates interface errors assumes the transition-layer model remains dominant; however, the error bounds for strong contrasts appear to omit the interaction term between numerical dispersion (which vanishes at CFL=1) and the discrete impedance mismatch, leaving the tightness of the strong-contrast estimate unclear when the Courant number is near unity.
minor comments (3)
  1. [Abstract] The abstract states that the analysis 'establishes benchmarks for validating structure-preserving discretizations,' yet the manuscript contains no direct numerical comparisons against alternative schemes (e.g., immersed-interface or high-order methods). This claim should be either substantiated or rephrased as a suggested future use.
  2. [§2] Notation for the two interface models (e.g., 'Model A' and 'Model B') is introduced without an explicit table summarizing the staggered locations of ε and μ for each; adding such a table would improve readability when the discrete coefficients are later compared.
  3. [§4] In the error-estimate derivations, the transition from the weak-contrast asymptotic to the strong-contrast bound is presented without an intermediate numerical check (e.g., a single plot of reflection coefficient versus contrast ratio at fixed Δx). Including one such verification plot would strengthen the transition-layer interpretation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the detailed, constructive comments. We address each major point below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (Derivation of discrete Fresnel coefficients): The effective boundary conditions are obtained by treating material parameters as pointwise assigned to staggered locations without averaging. This produces the claimed one-cell transition layer, but the manuscript does not demonstrate that the resulting error estimates remain valid (or provide conservative bounds) when common arithmetic or harmonic averaging is applied across the interface cell, which alters the effective impedance jump seen by the fields.

    Authors: Our analysis is performed for the standard Yee scheme with material parameters assigned directly to staggered locations (no averaging), which is one of the two common placements explicitly considered in the manuscript and which produces the one-cell transition layer. We agree that arithmetic or harmonic averaging alters the effective jump. In the revised manuscript we add a clarifying paragraph in §3 stating that the derived error estimates and transition-layer model apply to the non-averaged assignment; we also note that averaging would generally reduce the impedance contrast seen by the fields and therefore the interface error, but a quantitative treatment of averaged cases lies outside the present scope. revision: partial

  2. Referee: [§5] §5 (Courant-number dependence): The analysis of how the Courant number modulates interface errors assumes the transition-layer model remains dominant; however, the error bounds for strong contrasts appear to omit the interaction term between numerical dispersion (which vanishes at CFL=1) and the discrete impedance mismatch, leaving the tightness of the strong-contrast estimate unclear when the Courant number is near unity.

    Authors: The strong-contrast bounds in §5 are obtained from the exact discrete Fresnel coefficients that already embed the numerical wave number (hence the Courant number) through the 1-D dispersion relation. At CFL=1 the dispersion term vanishes identically in one dimension, so the remaining error is purely interfacial. To address the concern we will revise §5 to separate the dispersion contribution from the interface mismatch explicitly, insert the interaction term into the strong-contrast bound, and add a short paragraph showing that the bound remains valid (though not necessarily sharp) as CFL approaches unity. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from Yee equations and Fresnel theory

full rationale

The paper derives discrete reflection/transmission coefficients by applying the standard Yee update equations to two staggered-grid material placements, then formulates effective boundary conditions that emerge directly from those updates. The transition-layer interpretation is obtained by inspecting the resulting one-cell spread of the discontinuity; it is not presupposed or fitted. No parameters are tuned to data subsets and then relabeled as predictions, and no load-bearing step relies on self-citation chains or imported uniqueness theorems. The analysis remains within the classical Yee scheme and exact Fresnel formulas without reduction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard Yee FDTD discretization of Maxwell's equations for lossless media and classical Fresnel coefficients from prior literature.

axioms (1)
  • domain assumption The media are lossless, linear, homogeneous, and isotropic.
    Explicitly stated as the scope of the analysis in the abstract.
invented entities (1)
  • transition layer of one spatial step no independent evidence
    purpose: Models the effective spreading of the material discontinuity by the staggered grid
    Introduced as the key mechanism explaining systematic deviations from exact Fresnel coefficients.

pith-pipeline@v0.9.0 · 5595 in / 1385 out tokens · 30485 ms · 2026-05-16T21:32:39.545980+00:00 · methodology

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