{"id":"3afd94f6-8d56-4943-8062-24e75a231cbc","arxiv_id":"2505.18696","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Round-off error from double versus quadruple precision arithmetic changes the timing and apparent qualitative path of instability growth in the two-dimensional Taylor-Green vortex at Re=2000.","lead":"This study runs high-precision computer simulations of a classic 2D fluid flow, the Taylor-Green vortex, using double and quadruple precision arithmetic, and finds that the tiny round-off errors in the arithmetic change when and how the flow becomes unstable. The result suggests that machine precision can alter the apparent timing and structure of transition to turbulence in direct numerical simulations, which matters for reproducibility of CFD results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (3.4) is wrong: the assumed same-mode disturbance decays under the linearized VTE, so the claimed exponential growth rate 2/Re and the 'linear instability stage' are not established.","rationale":"The reader flagged the unquantified RK4 temporal error. My independent check reveals a more fundamental problem: the linear stability analysis itself is incorrect. A direct substitution into the VTE shows the single-mode disturbance decays (dF_d/dt = -2F_d/Re), the opposite of Eq. (3.3). The observed exponential growth in the simulations is therefore not the same-mode instability, and the claimed growth rate 2/Re is falsified by the paper's own timings (slope ≈0.07 vs 0.001). Because the 'receptivity phase' and 'linear instability stage' are defined relative to this flawed theory, the central claim about precision-delayed receptivity is not established. The empirical precision sensitivity may still be reproducible, but it is described within a wrong physical framework. This warrants rejection or at least a major rewrite with a corrected stability analysis; the current manuscript's conclusions overreach.","tokens_in":9471,"tokens_out":14797,"duration_ms":110565,"concrete_test":"Re-derive the O(ε) equation by substituting (3.1)-(3.2) into (2.1) and retaining linear terms in F_d; the advection contributions cancel exactly, yielding dF_d/dt = -2F_d/Re, the sign opposite to Eq. (3.3). If confirmed, compute the empirical slope of ln|ω_d,max| during the purported linear stage (e.g., between t=214 and 371 in the double-precision run) and compare with 2/Re=0.001; a slope larger by ~50x confirms that the observed growth is not the same-mode instability.","verdict_should_be":"REJECT","load_bearing_attack":"Substituting the ansatz (3.1)-(3.2) into the linearized VTE (2.1) gives dF_d/dt = -2F_d/Re, not +2F_d/Re, because u_m·∇ω_d + u_d·∇ω_m vanishes identically for the single-mode perturbation (the same cancellation makes the base-flow advection zero). Thus Eq. (3.4) cannot be the source of the growth observed in Fig. 2. Indeed, the reported times are inconsistent with the claimed rate: for double precision, |ω_d,max| grows from 10^-5 at t=214 to order 1 at t≈371, implying a natural-log slope ≈0.07, whereas 2/Re=0.001. The observed instability must arise from other Fourier modes (e.g., the saddle-point/nonmodal mechanism the authors invoke), not the mode in Eq. (3.4). Consequently the identification of phases AB and BC as 'receptivity' and 'linear instability' governed by this mode is unsupported, and the central claim that precision delays the receptivity route via this mechanism collapses. The precision difference may be a real numerical phenomenon, but the paper's theoretical framing and 'conclusively established' conclusion overreach.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9673,"tokens_out":8566,"duration_ms":71903,"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":[{"comment":"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.","section":"Section 3, Eqs. (3.3)-(3.4)"},{"comment":"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.","section":"Section 1 and Section 4"},{"comment":"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.","section":"Section 4, Figs. 2-4"},{"comment":"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.","section":"Section 4, phase AB"}],"minor_comments":[{"comment":"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.","section":"Section 3"},{"comment":"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.","section":"Fig. 3 caption"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Section 1"},{"comment":"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.","section":"Section 1"}],"recommendation":"major_revision","confidential_remarks":"The controlled numerical experiment is potentially salvageable, but the theoretical framing in Section 3 and the interpretation of the delay in Section 4 require substantial rework. I would not support acceptance without a corrected linear-stability derivation and a temporal convergence study."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know before reading: the numerical experiment is simple and the precision effect is probably real, but the paper's central linear-stability derivation is wrong. Substitute the ansatz from (3.1)-(3.2) into the linearized vorticity transport equation. The advection terms vanish because the disturbance has the same spatial structure as the base flow, leaving dF_d/dt = -2F_d/Re. That is decay, not growth. Eq (3.4) has the wrong sign. This is not a minor typo; it is the foundation for the \"linear instability stage\" and for the claim that round-off seeds a 2/Re exponential growth.\n\nWhat is genuinely new is the controlled double-vs-quadruple precision run of the 2D TGV benchmark, all other parameters fixed. That is a clean design, and the delay of the disturbance onset in quadruple precision is visible in Fig 2. The observation that round-off can serve as the receptivity seed is worth taking seriously.\n\nThe soft spots are not small. The measured slope between the 1e-5 level and saturation in the double-precision run is about 0.07 in log-linear units, not 0.001, so even empirically the 2/Re rate is not there. The \"receptivity phase\" is drawn by eye without a criterion, and the paper never quantifies the RK4 temporal truncation error at dt=0.025, so the claim that round-off is the only remaining error source is unsubstantiated. The \"qualitative difference\" between precisions is mostly a time delay; both runs go through the same four phases. The conclusions use \"conclusively established\" far beyond what the evidence and the corrected math support.\n\nMy take: the numerical fact may survive, but the paper needs a rewritten theory (likely nonmodal mechanisms), actual error budgets, and much softer claims. I would not cite it in its present form. If it lands in my in-box, I'd send it to referees because the observation is testable and the area matters, but any competent referee should catch the sign error and require major revision.","headline":"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.","tokens_in":10232,"tokens_out":3350,"would_cite":false,"duration_ms":29232,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["round-off error","Taylor-Green vortex","direct numerical simulation","receptivity","quadruple precision","pseudospectral method","vorticity dynamics","instability"],"falsifier":"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.","tokens_in":9218,"feed_emoji":"🌀","tokens_out":10070,"duration_ms":76808,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quadruple precision delays transition in identical vortex DNS runs","feed_subtitle":"Double vs quadruple arithmetic produces different receptivity, growth, and decay in the Taylor-Green vortex.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the exact decaying vortex solution and the perturbation series that the paper's linearized disturbance analysis extends.","marker":"Taylor & Green (1937)"},{"why":"Establishes the double-precision RK4-Fourier simulation of the 2D TGV problem that this paper repeats and extends to quadruple precision.","marker":"Sengupta et al. (2024)"},{"why":"Identifies the free saddle point as the onset location and gives the enstrophy-based receptivity theory used to interpret the early AB stage.","marker":"Sengupta et al. (2018)"},{"why":"Supplies the RK4 accuracy analysis used to justify the dt=0.025 time step and to argue that temporal error does not contaminate the comparison.","marker":"Sengupta et al. (2022)"},{"why":"Gives the enstrophy decay theory that the paper invokes for the final monotonic decay stage DE.","marker":"Doering & Gibbon (1995)"},{"why":"Supplies the Fourier spectral method and the 3/2-rule de-aliasing used to control truncation and aliasing errors.","marker":"Canuto et al. (1991)"},{"why":"Provides modal and non-modal stability results for the 2D TGV that the linear growth stage BC is compared against.","marker":"Gau & Hattori (2014)"}],"fun_headline_variants":["Round-off error steers turbulence route in Taylor-Green vortex","Quadruple precision shifts receptivity in vortex DNS","Floating-point precision alters vortex turbulence path","Round-off errors decide turbulence onset in vortex flow","Double vs quadruple arithmetic: different vortex decay routes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Round-off error steers turbulence route in Taylor-Green vortex","Quadruple precision shifts receptivity in vortex DNS","Floating-point precision alters vortex turbulence path","Round-off errors decide turbulence onset in vortex flow","Double vs quadruple arithmetic: different vortex decay routes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000724,"raw_usage":{"total_tokens":3219,"prompt_tokens":890,"completion_tokens":2329,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":2253}},"tokens_in":506,"tokens_out":2329,"duration_ms":13468,"temperature":1.0,"reasoning_tokens":2253,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:27:07.130714+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the double-precision RK4-Fourier simulation of the 2D TGV problem that this paper repeats and extends to quadruple precision."},{"cited_title":"Analysis of pseudo-spectral methods used for numerical simulation of turbulence","cited_arxiv_id":"2109.00255","evidence_quote":"Supplies the RK4 accuracy analysis used to justify the dt=0.025 time step and to argue that temporal error does not contaminate the comparison."},{"cited_title":"& Gibbon, J","cited_arxiv_id":null,"evidence_quote":"Gives the enstrophy decay theory that the paper invokes for the final monotonic decay stage DE."},{"cited_title":", Hussaini, M","cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier spectral method and the 3/2-rule de-aliasing used to control truncation and aliasing errors."},{"cited_title":"Fluid Dynamics Research 46 (3), 031410","cited_arxiv_id":null,"evidence_quote":"Provides modal and non-modal stability results for the 2D TGV that the linear growth stage BC is compared against."}],"review_version":1}