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REVIEW 4 major objections 5 minor 48 references

DNS and role of round-off error: Two-Dimensional Taylor-Green vortex problem

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In the two-dimensional Taylor-Green vortex, round-off error seeds the instability: quadruple precision delays receptivity and changes the route to turbulence and decay.

desk verdict The precision comparison is real, but the paper's linear growth rate has a sign error—Eq (3.4) predicts decay, so the central theoretical framing is wrong. read the letter →

arxiv 2505.18696 v1 pith:M3BWB6LA submitted 2025-05-24 physics.flu-dyn physics.comp-ph

V.K. Suman , T.K. Sengupta This is my paper · ORCID
classification physics.flu-dynphysics.comp-ph
keywords round-offerrorTaylor-Greenvortexdirectnumericalsimulationreceptivityquadrupleprecisionpseudospectralmethodvorticitydynamicsinstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that round-off error is not a passive numerical by-product but the actual seed of instability in the two-dimensional Taylor-Green vortex. By running the same Fourier-pseudospectral and RK4 simulation in double and quadruple precision with every other parameter identical, it reports that precision changes when receptivity begins, how fast the linear instability grows, and the qualitative structure of the disturbance field at the same amplitude. The authors also report a previously unidentified early receptivity phase during which precision-level background noise is converted into the log-linear growth predicted by the linearized vorticity equation. If this is right, the arithmetic precision of a DNS must be treated as part of the physics being simulated, not as a neutral background.

What carries the argument

The argument is carried by three ingredients working together. The first is the streamfunction-vorticity form of the 2D Navier-Stokes equations, discretized spatially with Fourier pseudospectral methods and advanced in time with fourth-order Runge-Kutta; this combination is meant to eliminate truncation, dispersion, and aliasing errors so that round-off is the only uncontrolled error. The second is the analytically known Taylor-Green equilibrium with the linearized disturbance law $\omega_d = \hat{F} e^{2t/Re} \sin x \sin y$, which predicts a log-linear growth stage against which the computed maximum disturbance vorticity is compared. The third is the precision contrast itself: two runs, double and quadruple, identical in every numerical parameter, serve as the experiment that isolates the effect of round-off. The onset location is the free saddle point at the cell center, and the named stages AB, BC, CD, and DE provide the vocabulary in which the precision effect is described.

What would settle it

Halve the time step to $dt = 0.0125$ while keeping double precision and the same de-aliasing; if the receptivity-phase onset shifts by an amount comparable to the delay caused by switching to quadruple precision at $dt = 0.025$, then temporal discretization error is contributing to the observed precision effect.

Watch

Extended reading notes

Core claim

The central claim is that the round-off error left by finite-precision arithmetic triggers receptivity in the 2D Taylor-Green vortex and that the level of precision therefore determines the instability route. Starting from the exact equilibrium $\psi_m = \sin x \sin y \, e^{-2t/Re}$, the linearized disturbance vorticity is $\omega_d = \hat{F} e^{2t/Re} \sin x \sin y$, and in the simulations this growth appears in four stages: receptivity (AB), linear growth (BC), nonlinear growth (CD), and enstrophy-driven decay (DE). When the same discretization is run in quadruple precision instead of double precision, the onset and duration of the receptivity phase shift significantly later in time, the disturbance fields differ in structure at matched amplitudes, and the decay phase is reached by a different route. The paper reads this as evidence that round-off error plays a singular role in the spatio-temporal vorticity dynamics, rather than acting as negligible noise.

Load-bearing premise

The load-bearing premise is that the Runge-Kutta time-stepping error at $dt = 0.025$ and the remaining aliasing are negligible or identical in the two precision runs, so that the only meaningful difference between them is the machine round-off; no convergence study is given to test this.

Editorial extensions

If this is right

  • Any DNS that does not control or report precision may be implicitly fixing a round-off seed; changing precision can move the transition time by a large factor.
  • Reported onset times for instability in the 2D TGV problem, and possibly in other receptivity problems, are precision-dependent and should be quoted with the precision used.
  • The newly identified receptivity phase AB must be included in analyses of transition, since it is the stage where background numerical noise is internalized into the flow's disturbance field.
  • The decay phase DE remains governed by the monotonic enstrophy decay of the 2D equations regardless of precision, so the precision effect is concentrated in the growth and receptivity stages.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural test the paper does not run is to vary the time step at fixed precision; if halving $dt$ shifts the receptivity onset as much as switching precision does, the temporal truncation error is entangled with round-off and the singular role needs qualification.
  • The paper's setup is unforced and uses no hyperviscosity, which suggests that earlier DNS studies using forcing or added dissipation may have masked precision effects; forced runs could be revisited in double versus quadruple precision to look for the same seed mechanism.
  • If the seed is just the amplitude of the initial round-off perturbation, then deliberately rounding the initial condition to different numbers of digits in the same double-precision code should reproduce the full range of onset times seen here; that would turn the qualitative claim into a quantitative scaling law.
  • A practical consequence for reproducibility is that DNS transition studies should specify precision and possibly rounding mode, since bit-level error paths could matter as much as the nominal number of digits.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper reports direct numerical simulations of the two-dimensional Taylor-Green vortex at Re = 2000 on a 128x128 grid, using a Fourier pseudospectral spatial discretization and RK4 time integration in both double and quadruple precision. The authors identify four stages in the evolution of the maximum disturbance vorticity (receptivity, linear instability, nonlinear saturation, and decay) and claim that quadruple precision delays the onset of the receptivity phase and produces a qualitatively different route to turbulence. They further claim that this establishes, for the first time, a singular role of round-off error in the spatio-temporal vorticity dynamics. The main evidence is the time history in Fig. 2 and the disturbance-vorticity field comparisons in Figs. 3 and 4.

Significance. The controlled precision comparison is a worthwhile idea: changing only precision while holding all numerical parameters fixed is a clean protocol for isolating round-off effects, and the use of a Fourier pseudospectral method removes spatial truncation error. If the claimed delay and structural differences were properly established, the results would be relevant to the DNS community, where double precision is the de facto standard. However, the present analysis contains a serious error in the linearized theory and does not rule out a trivial initial-amplitude explanation for the observed delay. The significance therefore depends on a substantial revision of the theoretical framing and on additional error-quantification experiments.

major comments (4)
  1. [Section 3, Eqs. (3.3)-(3.4)] The sign in the linearized disturbance equation is incorrect. Substituting the same-mode ansatz (3.1)-(3.2) into the linearized vorticity transport equation (2.1) gives dF_d/dt = -2F_d/Re, because u_m · grad(omega_d) + u_d · grad(omega_m) vanishes identically for this single-mode perturbation. Consequently Eq. (3.4) predicts exponential decay, not growth, and it cannot be the mechanism behind the stage BC identified in Fig. 2. The observed log-slope between DP2 (t=214, |omega_d,max|=1e-5) and DP3 (t=324, |omega_d,max| of order 0.3) is about 0.09, not 2/Re = 0.001. The 'linear instability stage' must be derived from a correct modal or nonmodal analysis of the time-dependent base flow, or removed from the interpretation.
  2. [Section 1 and Section 4] The assertion that 'the only major source of error ... is related to aliasing' ignores the temporal truncation error of the RK4 scheme. With dt = 0.025, no convergence study or error estimate is provided to show that the RK4 temporal error is smaller than double-precision round-off over the integration times considered. Without such a test, the double-versus-quadruple differences cannot be attributed solely to round-off; they could reflect a different mix of round-off and temporal truncation errors. The authors should provide a dt-refinement study (for example dt, dt/2, dt/4) in both precisions and report an error budget separating spatial, temporal, and round-off contributions.
  3. [Section 4, Figs. 2-4] The 'significant delay' of the quadruple run is not by itself evidence of a qualitatively different route, because it is largely consistent with the same growth process starting from a much smaller round-off seed. If both runs grow at the same rate g from initial amplitudes of order eps_d and eps_q, the time to reach a given amplitude is delayed by ln(eps_d/eps_q)/g. With eps_d/eps_q of order 1e-18 and the observed rate near 0.09, this offset is about 460 time units, comparable to the observed delay between DP2 (t=214) and QP2 (t=718). To support 'qualitatively different', the authors should show that growth rates, phase durations, or disturbance structures differ after normalizing for the initial round-off amplitude, or compare the runs at equal physical phase rather than at equal disturbance amplitude.
  4. [Section 4, phase AB] The identification of a 'receptivity phase' is not supported by any quantitative criterion. In the hydrodynamic-stability literature, receptivity refers to the conversion of external environmental disturbances into internal instability modes; here the disturbance is numerical round-off, and the AB stage may simply be the time required for numerical noise to exceed the precision floor and begin growing under the prevailing linear dynamics. The paper should define a measurable diagnostic (for example, spectral content, projection onto a growing mode, or spatial correlation) that distinguishes AB from a numerical transient, and should justify the use of the term 'receptivity' for this process.
minor comments (5)
  1. [Section 3] The expression for omega_{d,epsilon} after Eq. (3.4) is garbled in the manuscript; it should be typeset as a clear expansion of the exponential and the notation F_d versus hat{F} should be unified.
  2. [Fig. 3 caption] The caption says 'same tolerance level of 10^-5'; this should refer to |omega_d,max| = 10^-5, not to Log(|omega_d,max|), to avoid confusion with the vertical axis of Fig. 2.
  3. [References] The reference 'Sengupta et al. (2022)' is listed with 'arXiv:2109.00255 10, 9-24', which appears to be an incomplete or incorrect journal citation; please verify and complete the bibliographic details.
  4. [Section 1] The sentence 'Buaria et al. (2020), who used 122883 periodic grids' appears to contain a typo; it should likely read 12288^3, and should be corrected.
  5. [Section 1] The phrase 'the truncation error is absent' should be qualified as applying to the spatial discretization; the temporal discretization still introduces truncation error, as noted in the major comments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the precision-controlled DNS comparison is self-contained, and no fitted parameter is renamed as a prediction. The theoretical Eq. (3.4) contains a likely sign error and is inconsistent with the reported growth slope, but that is a correctness issue, not an input-output circularity.

full rationale

The paper's central evidence is a controlled precision experiment: double and quadruple precision runs use identical grids, Reynolds number, time step, and numerical method, so the observed differences in onset and evolution of the disturbance are directly attributable to precision rather than to a fitted parameter. The disturbance growth formula in Eq. (3.4) is presented as a linearized prediction from the vorticity transport equation; it is not fitted to the DNS output, so it is not a 'prediction' that reduces to its inputs by construction. The same-mode ansatz (3.2) is a standard modal assumption, and even though substituting it into the linearized VTE actually gives dF_d/dt = -2F_d/Re (the advective terms cancel identically and diffusion decays the mode), this is a mathematical correctness concern, not circularity: the growth rate would have been an independent analytical result if the algebra were correct. Section 4 also labels the AB portion as a 'receptivity phase'; this is a descriptive identification of the initial noise-growth interval, not a self-definitional derivation. The manuscript does rely on several citations to the authors' prior work, e.g., 'A time step of 0.025 is chosen which gives very good accuracy according to analysis by Sengupta et al. (2022)', and on assertions that truncation error is negligible, but the load-bearing conclusion about round-off effects is supported by the paper's own controlled comparison rather than by those citations alone. No uniqueness theorem is imported, and no known result is merely renamed as organization. The main weaknesses are the apparent sign error in Eq. (3.4), the mismatch between the reported observed slope (about 0.07 in natural-log units) and 2/Re = 0.001, and the lack of a temporal-convergence study to support the claim that round-off is the sole uncontrolled error; these are correctness and evidence-strength limitations, not circular reasoning. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The ledger shows the paper relies on a standard linear stability result, a controlled precision experiment, and several unquantified numerical assumptions (temporal error, de-aliasing, Gaussian noise). No fitted parameters appear in the central derivation; the free parameters are the chosen simulation configuration.

free parameters (4)
  • Reynolds number Re = 2000
    The Reynolds number of the problem, chosen as the benchmark case, not fitted to data.
  • Time step dt = 0.025
    Time step chosen based on prior analysis by Sengupta et al. (2022), not fitted to the present data.
  • Grid size N = 128x128
    Number of Fourier modes / grid points, chosen to resolve the 2x2 cell TGV at Re=2000, asserted adequate without a resolution study.
  • Vortex cell configuration = 2x2 cells
    Configuration selected to produce free and constrained saddle points; a prior 4x4 cell study (Sengupta et al. 2024) exists, so this is a choice, not a fitted parameter.
assumptions (4)
  • domain assumption The disturbance amplitude epsilon is small enough for linearization of the vorticity transport equation.
    Substitution of psi=psi_m+eps psi_d and omega=omega_m+eps omega_d into the VTE and keeping O(eps) terms yields Eq. (3.3); this requires the disturbance to remain small, which is plausible for early times but unverified at later times.
  • standard math Round-off errors are normally distributed via the central limit theorem.
    Section 4 invokes the CLT to treat round-off as Gaussian noise; this is a standard but not proven assumption for this particular simulation, as the errors may not be independent or identically distributed.
  • domain assumption The 2-rule zero-padding completely removes aliasing error.
    Section 1 states that the 2-rule proposed in Sengupta (2004) ensures complete removal of aliasing; this is accepted from the cited reference without proof in the present paper.
  • ad hoc to paper The RK4 temporal truncation error is negligible compared to round-off error.
    The paper asserts all errors except round-off are controlled, but RK4 with dt=0.025 has a finite temporal error that is not quantified; this is a load-bearing assumption for attributing differences to precision alone.

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Cite this review

Pith. "Pith review of DNS and role of round-off error: Two-Dimensional Taylor-Green vortex problem." pith.science (2026). https://pith.science/paper/M3BWB6LA

@misc{pith2026250518696,
  author       = {Pith},
  title        = {Pith review of: DNS and role of round-off error: Two-Dimensional Taylor-Green vortex problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M3BWB6LA}},
  note         = {Machine review of arXiv:2505.18696}
}
read the original abstract

The role of round-off errors on the receptivity and instability of fluid flows are conclusively established for the first time using high accuracy simulations of the benchmark two-dimensional (2D) Taylor-Green vortex problem using double and quadruple precisions. Employing the fourth order Runge-Kutta (RK4) method for temporal discretization and Fourier pseudospectral method for spatial discretization enables unprecedented accuracy necessary for controlling all forms of errors, except the remaining round-off error. Results clearly show that adopting quadruple precision results in a qualitatively different receptivity route to turbulence and its subsequent decay compared to double precision. Another important observation is the identification of the receptivity phase which has never been reported before. Present study not only establishes the singular role of round-off errors but also has potential ramifications on receptivity and instability of flows due to precision of simulation.

Figures

Figures reproduced from arXiv: 2505.18696 by the authors.

Figure 1
Figure 1. Free (FSP) and constrained (CSP) saddle points in the initial solution of the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Phases of disturbance growth and the effect of precision for the 2D TGV [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Comparison of the disturbance vorticity fields between double and quadruple [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparison of the disturbance vorticity fields between double and quadruple [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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    Diffusion in inhomogeneous flows: Unique equilibrium state in an internal flow – Sengupta, T. K., Singh, H., Bhaumik, S. and Roy Chowdhury, R. Comput. Fluids, 88, 440–451 (2013) Here, we report for the first time the effects of computing precision on the dynamics of the flow a...

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    This probability distribution can be poorly approximated by any normal distribution

    It has been noted Ames that when a number is rounded to m^ th decimal place, the error is never larger in magnitude than one half unit in the place of the m^ th digit of the rounded number, and the probability density function has a constant value, p = 1 2| |_ max for | | < | ...

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Reviewed August 7, 2026 · model on record in the stance chip above.