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REVIEW 2 major objections 4 minor 30 references

Setting the Courant number equal to the refractive index removes numerical dispersion from 2D FDTD wave simulations and works for rarefied and left-handed media.

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2026-07-10 22:57 UTC pith:NLZGVTLH

load-bearing objection Solid applied note that Sc = nr cancels monochromatic phase error and works for nr < 1 and left-handed media; core algebra is textbook, stability and broadband claims are only partially supported. the 2 major comments →

arxiv 2607.06707 v1 pith:NLZGVTLH submitted 2026-07-07 physics.comp-ph math-phmath.MPphysics.optics

Impact of Courant number on the results of numerical simulating of signal propagation in non-dispersive homogeneous media

P. A. Makarov , R. N. Skandakov , V. A. Ustyugov , V. I. Shcheglov This is my paper
classification physics.comp-ph math-phmath.MPphysics.optics
keywords FDTDCourant numbernumerical dispersionYee gridnondispersive medianegative refractionleft-handed mediaphase velocity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Standard Yee FDTD calculations of electromagnetic pulses in homogeneous nondispersive media produce artificial numerical dispersion whenever the Courant number is left at the vacuum value Sc = 1. This paper derives the exact numerical dispersion relation on the staggered grid and shows that the phase velocity on the grid becomes identical to the physical phase velocity for every wavelength discretization once Sc is set equal to the relative refractive index nr = √(εr μr). That single choice therefore eliminates numerical dispersion, restores correct pulse shapes, and simultaneously stabilizes the update scheme for media optically thinner than vacuum as well as for left-handed media with simultaneously negative permittivity and permeability. The result is presented as four statements that fix both the optimal Courant number and the hard stability boundary Sc ≤ nr.

Core claim

For any real relative permittivity and permeability belonging to the paper's domain of definition, the choice Sc = √(εr μr) makes the Yee-grid phase velocity coincide exactly with the continuous-space phase velocity independently of the wavelength sampling Nλ, thereby removing numerical dispersion; the same choice is the unique optimal value that keeps the discrete algorithm stable and extends it to nr < 1 and to left-handed media.

What carries the argument

The closed-form numerical dispersion relation (Proposition 1) obtained by substituting plane-wave modes into the staggered Yee updates, which yields the phase-velocity ratio (27) whose only free parameter that can be tuned to unity is the Courant number Sc.

Load-bearing premise

The claim that the scheme always diverges when Sc exceeds nr rests only on numerical illustrations and a physical-level argument, not a complete discrete stability proof.

What would settle it

Run the identical Yee update (2)–(5) with Sc slightly larger than nr on a sequence of successively refined grids; if the solution remains bounded for some meshes while nr is held fixed, the asserted unique optimality of Sc = nr fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript studies numerical dispersion of the 1D Yee FDTD scheme for nondispersive homogeneous media with refractive index nr = √(εr μr) ≠ 1. Using shift operators, it derives the discrete dispersion relation (17) and the monochromatic phase-velocity ratio (27). Proposition 3 shows that the algebraic choice Sc = nr makes ˜cp/cp = 1 for every wavelength discretization Nλ, eliminating monochromatic phase error. Numerical examples with Gaussian and Ricker pulses (Figs. 5, 7) and a left-handed medium (Fig. 8) illustrate that the same choice also works for broadband packets and for media with nr < 1 or εr = μr = −1. Proposition 4 asserts that the scheme (2)–(5) diverges for Sc > nr, supported by a numerical illustration (Fig. 6). The authors conclude that Sc = nr is the unique optimal Courant number for the class of media considered.

Significance. If the claims hold, the paper supplies a simple, parameter-free rule that removes the usual numerical-dispersion limitation of Yee FDTD in homogeneous media and simultaneously extends a standard update scheme to optically less dense and left-handed media. The monochromatic derivation (Lemma 1, Propositions 1–3) is clean and the algebraic cancellation Sc = nr is exact, not fitted. The numerical illustrations for nr < 1 and for negative-index media are of practical interest and are rarely shown with ordinary Yee updates. These strengths make the work potentially useful for FDTD practitioners, provided the broadband and stability gaps identified below are closed.

major comments (2)
  1. [Proposition 3, Corollary 4, Abstract] Proposition 3 and the subsequent claim in Corollary 4 / the abstract that Sc = nr “eliminates the numerical dispersion … for signals of any shape and spectral composition” rest only on monochromatic cancellation of (27). Because the discrete dispersion relation (17) is nonlinear in frequency, exact phase-velocity matching at every ω does not automatically guarantee that a finite-bandwidth packet retains its shape (group-velocity or amplitude errors may remain). The paper supplies no analytic argument that the full discrete transfer function becomes nondispersive for a continuum of frequencies, nor any quantitative shape-error metric (e.g., L2 residual versus propagation distance). The visual agreement of two particular waveforms (Gaussian, Ricker) in Figs. 5 and 7 is insufficient to support the general claim.
  2. [Proposition 4, Fig. 6, Corollary 4] Proposition 4 asserts that the update scheme (2)–(5) diverges whenever Sc > nr and is therefore the unique stability boundary that makes Sc = nr optimal. The only support offered is a single numerical illustration (Fig. 6) and an argument “at the physical level of rigor.” No von Neumann, matrix, or energy stability analysis is given. Without such a proof the forbidden region B in Fig. 4 and the uniqueness statement of Corollary 4 remain incompletely justified; a mesh-dependent or source-dependent stability boundary would weaken the optimality claim.
minor comments (4)
  1. [Abstract, Introduction, Eq. (1)] The paper is written for a 1-D spatial grid (one spatial + one temporal dimension) yet repeatedly refers to the “2D case.” Standard FDTD terminology reserves “2D” for two spatial dimensions; the wording should be corrected to avoid confusion.
  2. [Remark 11, Fig. 7] Remark 11 and Fig. 7 document a spurious backward pulse Pb whose amplitude grows as nr o 0. The phenomenon is left unexplained and unquantified; a short discussion of its origin (TF/SF source, floating-point, or analytic inconsistency) and a bound on its relative amplitude would strengthen the practical utility of the method.
  3. [Figs. 1, 2, 5] Several figures (especially Figs. 1, 2, 5) lack axis labels or legends that make the Courant number and refractive index immediately readable; adding them would improve reproducibility.
  4. [Some open questions] The open questions listed at the end are valuable, but the first two (accuracy for nr < 1 and origin of Pb) are already partially answerable from the material in the paper and could be addressed briefly rather than left entirely open.

Circularity Check

1 steps flagged

Algebraic cancellation of monochromatic phase error is self-contained; only minor non-load-bearing self-citation to prior scheme definition.

specific steps
  1. self citation load bearing [Remark 1 and comparison of pulses P1/P3 vs P2/P4 (Introduction / Sec. 1)]
    "the pulses P1 and P3 can be considered a correct numerical solution of the problem in the sense of Definition 2 given in [18]. Proceeding from this, the pulses P1 and P3 can be considered “reference” ones"

    Correctness of the vacuum reference solutions is imported wholesale from the authors’ own prior paper rather than re-derived or independently verified here; however the import is used only for visual comparison and is not required for the algebraic identity of Proposition 3, so the circularity is minor and non-load-bearing for the central claim.

full rationale

The core derivation chain is independent and non-circular. Lemma 1 and Proposition 1 obtain the Yee dispersion relation (17) directly from the discrete Ampere/Faraday update equations via shift operators, without external input or fitting. Proposition 2 then produces the phase-velocity ratio (27) by elementary algebra from (17) and the continuous relation (25). Proposition 3 is the direct substitution Sc = nr into (27), which yields ˜cp/cp = 1 identically for every Nλ; this is an algebraic identity, not a fitted or self-defined prediction. Corollary 4’s uniqueness claim rests on that identity plus the numerical observation of divergence for Sc > nr (Proposition 4 / Fig. 6), which is weak evidence but not circular. Self-citations to the authors’ prior work [18] supply only the concrete update stencil (2)–(5), the TF/SF source, and Definition 2 of “correctness”; they are not invoked as the content of the dispersion identity itself. No parameters are fitted to data and then re-presented as predictions, no uniqueness theorem is imported from the same authors to forbid alternatives, and no known empirical pattern is merely renamed. The monochromatic-to-broadband extension (Corollary 4, abstract) is an overclaim of scope rather than a circular reduction. Hence only a residual score of 1 for the non-load-bearing self-citation of the scheme.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claim rests on the standard Yee discretization of Maxwell’s equations, a monochromatic plane-wave ansatz on the grid, and the modeling restriction to homogeneous nondispersive media with real εr, μr (and εr μr > 0 for the left-handed extension). No free parameters are fitted to data; Sc and Nλ are chosen discretization controls. No new physical entities are introduced—the “magic” Courant number is only a name for Sc = nr.

axioms (4)
  • domain assumption Yee staggered finite-difference discretization of Faraday and Ampere laws is a consistent second-order approximation to continuous Maxwell equations in homogeneous media.
    Used from the Introduction through §2 as the starting point for operators (6)–(11) and the dispersion derivation.
  • domain assumption Material parameters εr, μr are real constants (nondispersive, homogeneous); domain (34) and later εr μr > 0 for left-handed media.
    Stated in the abstract, §2 domain of definition, and §4 limits; excludes the strongly attenuating εr μr < 0 case.
  • standard math Plane monochromatic wave ansatz (12)–(13) on the discrete grid yields the numerical dispersion relation via substitution into the update equations.
    Lemma 1 and Proposition 1; classical von Neumann-style analysis.
  • ad hoc to paper When Sc > nr the scheme is unstable (self-excitation destroys the solution).
    Proposition 4 is supported only by numerical examples (Fig. 6) and “physical level of rigor,” not a full discrete stability theorem.

pith-pipeline@v1.1.0-grok45 · 17976 in / 3185 out tokens · 41981 ms · 2026-07-10T22:57:54.702365+00:00 · methodology

0 comments
read the original abstract

The paper is devoted to the study of the connection between the numerical dispersion arising in FDTD modeling of electromagnetic signal propagation in nondispersive homogeneous media optically different from vacuum and the Courant number in the 2D case. The main results are formulated in the form of four statements, as well as a number of corollaries and remarks that determine the nature of the numerical dispersion, the optimal value of the Courant number and the limitations of the method. It is proved that the optimal choice of the Courant number eliminates the numerical dispersion and extends the capabilities of the developed numerical algorithm to media, which refractive index lesser than refractive index of vacuum, as well as media with negative refraction.

Figures

Figures reproduced from arXiv: 2607.06707 by P. A. Makarov, R. N. Skandakov, V. A. Ustyugov, V. I. Shcheglov.

Figure 1
Figure 1. Figure 1: Simulating of the propagation of Gaussian form pulses in vacuum [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulating of pulse propagation in the form of a Ricker wavelet in vacuum [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dependence of the phase velocity ratio, determined according to the FDTD method [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: pNλ, Scq-diagram of Yee grid bandwidths when nr “ 100. medium nr “ 100. The regions of the stopbands tAiu N i“1 , corresponding to the condition |φpNλ, Sc, nrq| ą 1, are shown in this diagram by a monotone gray shading. The total number of such regions N (the first two of them are marked in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simulating of Gaussian pulse propagation in media optically denser [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time dynamics of the numerical solution destruction obtained during computation [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Dynamics of forward Pf and backward Pb pulse forming in simulating the propagation of a Gaussian-shaped signal in media optically less dense than vacuum under optimal choice of the Courant number. The dashed-dotted curve 1 corresponds to the case of Sc “ nr “ 10´1 , the solid line 2 — Sc “ nr “ 10´2 . where the energy flux carried by the wave, and determined by the vector S, is equal to [20–23, 26] S “ c 4… view at source ↗
Figure 8
Figure 8. Figure 8: Instantaneous snapshots of forward (bottom half) and backward (top half) waves [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

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Reference graph

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