{"id":"36fd8cb4-32e8-4a45-a58f-d6f403d379b7","arxiv_id":"2502.08116","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A pressure-based finite difference method with simultaneous relaxation of velocity, pressure, density, and internal energy achieves near-roundoff discrete conservation of momentum and total energy for low Mach number compressible flows.","lead":"This paper proposes a numerical method for low Mach number compressible flows that simultaneously corrects velocity, pressure, density, and internal energy while preserving discrete conservation of momentum and total energy. It matters because it combines existing conservative schemes with a new relaxation step to enable stable, energy-conserving simulation of flows where density varies in space and time.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Roundoff-level total energy conservation depends on unstated convergence of the simultaneous relaxation loop; no iteration count or tolerance is reported.","rationale":"The reader's weakest assumption correctly identifies the simultaneous relaxation loop and the unstated iteration count as the load-bearing condition for the strongest claim. My own reading confirms this: the discrete total-energy conservation shown in Fig. 10 is not derived in the paper but is asserted on the basis of the fully conservative scheme. An algebraic check of Eqs. (3.23b) and (3.23c) shows that the pressure-work and kinetic-energy corrections cancel only when the linearized quantities used in the right-hand sides match the final converged values, i.e., when both the inner l-loop and outer m-loop have converged. Since the paper reports no iteration counts, tolerances, or convergence diagnostics, the roundoff-level conservation claim is not independently verifiable. The entropy overclaim (abstract and conclusion say entropy is conserved, while Section 4.2 reports |ερs| ~ 1e-6 and explicitly states the discrete entropy conservation property worsens) is also a genuine defect, but it is a claim-revision issue rather than a threat to the main technical result. The convergence issue strikes at the heart of the claimed advantage—roundoff-level total energy conservation—and is therefore the single most load-bearing concern. A concrete rerun with varied iteration counts would settle whether the reported figures are robust or an artifact of an adequately converged but unreported iteration count.","tokens_in":33792,"tokens_out":5583,"duration_ms":68197,"concrete_test":"Rerun the Section 4.2 periodic inviscid compressible flow (11^3 grid, Δt/(L/U0)=0.002, Ma=0.2) with explicit inner relaxation counts l_max = 1, 2, 5, 10 and outer Newton counts m_max = 1, 2, 3, 5, and report |ερE| at t/(L/U0)=10. If |ερE| remains ~1e-15 already for l_max=m_max=1, the concern is resolved; if it degrades with lower iteration counts, the paper must report the iteration count used for Fig. 10 and demonstrate convergence. Also report the convergence criterion used for 'Newton iteration completed' in Step 7.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (Section 4.2, Fig. 10) is that |ερE| stays near 1e-15 independent of Δt when kinetic and internal energies are stored at time level n+1. This discrete conservation is only guaranteed if the outer Newton loop (Steps 2-7) and the inner simultaneous relaxation loop (Eqs. 3.23a-e) are iterated to convergence, because the scheme's conservation identity relies on replacing ρ^{n+1,m+1} by ρ^{n+1,m} and û^{n+1,m+1} by û^{n+1,m} (Section 3.2). The authors state 'We repeat the calculation up to a predetermined iteration number' but never report that number, any tolerance, or a convergence study. Stopping early breaks the discrete total-energy balance: summing Eqs. (3.23b) and (3.23c) leaves an error proportional to Δp^l (∇·u^m − ∇·u^*), which vanishes only at full convergence. Thus the headline conservation result, as reported, is not reproducible and may not hold in general.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a pressure-based finite-difference method for low Mach number compressible flows, built on the fully conservative scheme of Morinishi and the semi-implicit acoustic-treatment ideas of Wall, Pierce, and Moin. The key algorithmic novelty is a simultaneous relaxation loop in which velocity, pressure, internal energy, and density are corrected together, with density obtained from the equation of state. The method is tested on sound-wave propagation, three-dimensional periodic inviscid compressible flow, decaying compressible isotropic turbulence, Taylor-Green vortex flows, natural convection in a cavity, and a three-dimensional Taylor decaying vortex. The central claims are second-order accuracy and discrete conservation of momentum and total energy at roundoff level when kinetic and internal energies are stored at the same time level n+1, with a weaker entropy conservation result.","tokens_in":33900,"tokens_out":5587,"duration_ms":50516,"significance":"If the central conservation claims hold, the method would be a practically useful pressure-based low Mach number solver with excellent long-time conservation properties, validated against several external references including the exact three-dimensional Navier-Stokes solution of Antuono, the DNS of Honein and Moin, the wall-bounded natural-convection benchmark of Vierendeels et al., and the Taylor-Green reference data. The broad benchmark coverage is a genuine strength, as is the use of externally defined reference solutions rather than self-consistency checks. The technical novelty is incremental, however: the roundoff-level total-energy conservation at time level n+1 is essentially a property of the underlying Morinishi fully conservative scheme, and the paper's own Fig. 10 confirms that Morinishi's method already achieves this behavior. The contribution of the simultaneous relaxation loop is stability and practical iteration behavior, but that contribution is not quantified anywhere in the manuscript.","major_comments":[{"comment":"The text in Section 3.2 says that the Poisson equation for pressure should be solved after correcting the velocity, and that the present method 'does not require the Poisson equation for the pressure to be solved.' This wording is confusing because the Helmholtz equation for Δp is derived from the mass conservation constraint and serves as the pressure-correction equation. Please clarify the distinction between a pure Poisson solve and the Helmholtz form used here.","section":"§3.2, Eqs. (3.15), (3.23), and Steps 2–7; Figs. 9–10"}],"minor_comments":[{"comment":"The third definition in Eq. (3.30) repeats ερu instead of defining ερw; this is a typo that should be corrected.","section":"§3.4, Eq. (3.30)"},{"comment":"The sentence 'The present numerical method does not require the Poisson equation for the pressure to be solved' is misleading because Eqs. (3.19) and (3.23a) solve a Helmholtz equation for the pressure correction. Please rephrase to say that the pressure-correction Helmholtz equation is solved directly.","section":"§3.2"},{"comment":"There is a typo in 'hihgly SMAC method'; it should read 'highly SMAC method.'","section":"§3.2"},{"comment":"The conclusion that the present method and the Wall et al. variant show 'no difference in characteristics' in Fig. 8 is contradicted by the large difference in total-energy conservation error in Fig. 9; the sentence should be qualified.","section":"§4.2"},{"comment":"The Eckert number Ec is used in Eq. (2.11) but defined only later in the text; the definition should be moved before first use.","section":"§2.1"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is well grounded: Section 3.2 never reports iteration counts or tolerances for the simultaneous relaxation loop, and this is exactly what prevents the roundoff-level conservation claims from being reproducible. I recommend major revision rather than rejection because the method is benchmarked against several external references and the missing information can be supplied with a convergence study. The novelty is incremental with respect to Morinishi and Wall et al., so the editor may wish to weigh whether the simultaneous relaxation contribution is sufficiently demonstrated once the requested iteration data are added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new piece here is the simultaneous relaxation loop, Eqs. (3.23a-e), which corrects density and internal energy together with velocity and pressure. Wall et al. only correct velocity/pressure, and Morinishi's fully conservative scheme has no pressure-correction step of this kind. Combining Wall's Helmholtz pressure correction with Morinishi's square-root density weighted interpolation is a sensible engineering move, and the benchmark set is unusually broad: sound-wave dispersion matches theory, the Antuono exact solution, Honein and Moin DNS, Vierendeels cavity results, and Taylor-Green all land well. The convergence plots show clean slope -2, and momentum and total energy sit at roundoff in the periodic inviscid cases. The authors earn credit for testing nonuniform grids too.\n\nThe soft spots are real but not fatal. First, the abstract and conclusion claim discrete entropy conservation, but the body shows |ερs| ~ 1e-6 to 1e-3 and explicitly says the entropy conservation property worsens. That is an overclaim and should be corrected. Second, the simultaneous relaxation loop is run to a \"predetermined iteration number\" that is never reported. The roundoff-level total energy conservation depends on the Newton/simultaneous relaxation linearization being sufficiently converged; without the iteration count or a convergence study, the headline conservation result is not reproducible as reported. This is a genuine gap, and the stress-test note is right to flag it. I would also note that the roundoff-level total energy behavior at internal energy time level n+1 is largely inherited from Morinishi's scheme, so the new contribution is the density/internal-energy relaxation and its stability effect, not the conservation property itself. No code or data are provided, which limits independent verification but is common for this type of paper.\n\nThis is a legitimate incremental advance, not a breakthrough. For someone building a pressure-based low-Mach solver, the method is directly useful, and the numerical comparisons are thorough enough to deserve referee time. The paper needs a revised abstract/conclusion and an explicit statement of the relaxation iteration count and tolerance before acceptance.","headline":"A competent incremental advance in pressure-based low-Mach solvers, with an unverified iteration count and an entropy-conservation overclaim that need fixing before it is publishable.","tokens_in":34544,"tokens_out":1160,"would_cite":true,"duration_ms":13428,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M06","76M20","76N15"],"pacs":["47.11.-j","47.11.Bc","47.40.-x"],"model":"deepseek-v4-flash","headline":"A simultaneous-relaxation finite difference method solves low Mach number compressible flows without low Mach number approximations, conserving momentum and total energy at roundoff level.","keywords":["low Mach number","compressible flow","conservative finite difference","simultaneous relaxation","total energy conservation","pressure-based method","internal energy equation","density variation"],"falsifier":"Rerun the three-dimensional periodic inviscid test of Section 4.2 with the same grids and time steps but with the simultaneous relaxation limited to 1, 2, 5, 10, and 100 iterations, and record $|\\varepsilon_{\\rho E}|$ at $t/(L/U_0)=10$. If roundoff-level conservation appears only beyond some iteration count, or fails to appear at any finite count, then the conservation claim depends on the loop being run to convergence rather than being an intrinsic property of the discretization. A second check is the entropy error, which the paper reports at $10^{-6}$ rather than roundoff; grid-refinement and iteration-count studies of $|\\varepsilon_{\\rho s}|$ would show whether entropy conservation improves or saturates.","tokens_in":33475,"feed_emoji":"🌀","tokens_out":6857,"duration_ms":53235,"temperature":0.7,"pith_summary":"The paper proposes a numerical method for low Mach number compressible flows that avoids low Mach number approximations and still achieves discrete conservation of momentum, total energy, and entropy. The method relaxes velocity, pressure, density, and internal energy simultaneously inside each Newton step, using a fully conservative finite difference scheme on a staggered grid in space and time. Its central finding is that when kinetic and internal energies are stored at the same time level $n+1$, total energy is conserved at roundoff level (about $10^{-15}$) and momentum to about $10^{-17}$ in a three-dimensional periodic inviscid test. A sympathetic reader would care because this could give long-time low Mach number simulations without the drift and instability that often accompany approximate low Mach treatments.","feed_headline":"Low-Mach flow solver conserves total energy to roundoff","feed_subtitle":"Relaxing velocity, pressure, density, and internal energy together keeps momentum and energy exact.","key_machinery":"The central object is the simultaneous relaxation loop in Eqs. (3.23a)–(3.23e): inside each Newton iteration, a Helmholtz equation for the pressure correction replaces the usual Poisson equation (adding the density derivative $\\partial\\rho/\\partial p|_e$), and then velocity, pressure, internal energy, and density are corrected together, with density updated through the equation of state $\\rho = (\\kappa Ma^2 p + 1)/e$. Around this loop sits a fully conservative finite difference scheme with square-root density weighted interpolation of velocity and internal energy, a spatiotemporal staggered grid, and the implicit midpoint rule. The identity that carries the argument is the discrete placement of kinetic and internal energy at the same temporal level $n+1$, which makes the total energy constant at roundoff level; placing internal energy at $n+3/2$ instead degrades the total energy error to about $10^{-7}$.","core_discovery":"On its own terms, the paper claims that a pressure-based, fully conservative finite difference scheme can solve the compressible Navier–Stokes equations at low Mach numbers without invoking low Mach number, Boussinesq, or incompressibility approximations, and that this scheme discretely conserves the total amounts of momentum, total energy, and entropy. The key numerical finding is that placing kinetic and internal energies at the same temporal level $n+1$ makes the discrete total energy constant at roundoff level ($|\\varepsilon_{\\rho E}| \\approx 10^{-15}$), whereas the earlier approach of Wall et al. with internal energy at $n+3/2$ gives $|\\varepsilon_{\\rho E}| \\sim 10^{-7}$ on the same test. The scheme also reproduces sound wave amplitude and frequency, turbulence statistics, Taylor–Green vortex decay, natural convection in a cavity, and three-dimensional Taylor vortex decay, supporting its accuracy across inviscid, viscous, compressible, and incompressible regimes.","pith_inferences":["Because the scheme avoids low Mach number approximation, the same discretization could in principle handle flows that transition between compressible and incompressible behavior, such as combustion or high-temperature heat sources, without model switching; the paper does not test this directly.","The reported entropy conservation at $10^{-6}$ is orders of magnitude weaker than momentum and total energy conservation, suggesting that entropy is only approximately conserved; entropy-based diagnostics should be used with caution in long integrations.","The missing iteration-count reporting makes a reproducibility check natural: fixing a small, fixed number of relaxation iterations would determine whether roundoff-level energy conservation is robust to under-relaxation, which matters for production codes.","The same simultaneous relaxation idea could be applied to other tightly coupled systems, such as magnetohydrodynamic or reacting flows, where pressure, density, and energy updates interact strongly; the paper does not address these."],"forward_implications":["Low Mach number flows with large temperature differences, where density varies and incompressibility fails, can be computed with the same solver as compressible and incompressible regimes, without switching models.","Long-time simulations of periodic inviscid flows will not accumulate momentum or total-energy drift, since those quantities are conserved at roundoff level on both uniform and nonuniform grids.","The scheme remains stable at Courant numbers above 1 (up to CFL 8 for sound waves) because the pressure is treated implicitly and the pressure correction takes Helmholtz form.","The method extends to viscous compressible turbulence and natural convection, matching benchmark dissipation rates and Nusselt numbers while using fewer grid points than some earlier work."],"supporting_citations":[{"why":"Supplies the semi-implicit pressure treatment and double-time pressure interpolation that the method extends, and provides the baseline for sound wave amplitude and frequency comparisons.","marker":"(Wall et al., 2002)"},{"why":"Supplies the fully conservative finite difference scheme with square-root density weighted interpolation and the n+1 internal energy time level that yields roundoff-level total energy conservation, plus baseline turbulence results.","marker":"(Morinishi, 2009, 2010)"},{"why":"Provides the barely implicit correction baseline whose sound wave amplitude attenuation is compared against the present method.","marker":"(Patnaik et al., 1987)"},{"why":"Justifies using the internal energy equation instead of the total energy equation for stability and provides direct numerical simulation baselines for decaying compressible turbulence.","marker":"(Honein and Moin, 2004)"},{"why":"Provides benchmark solutions for natural convection in a cavity with large horizontal temperature differences, against which the present pressure and Nusselt numbers are compared.","marker":"(Vierendeels et al., 2003)"},{"why":"Provides the tri-periodic fully three-dimensional analytic Navier–Stokes solution used to verify the three-dimensional Taylor decaying vortex calculations.","marker":"(Antuono, 2020)"},{"why":"Supplies the nondimensionalization that makes the equations reduce to incompressible form as the Mach number approaches zero, a property the present method preserves.","marker":"(Bijl and Wesseling, 1998)"}],"fun_headline_variants":["Simultaneous relaxation yields exact energy conservation","Low-Mach solver conserves energy to machine precision","Fully conservative scheme for low-Mach flows","All variables relaxed together for exact energy","Energy conservation to roundoff without low-Mach tricks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the simultaneous relaxation loop is iterated enough times that the linearizations used in the Newton loop (replacing $\\rho^{n+1,m+1}$ by $\\rho^{n+1,m}$ and $\\hat{u}^{n+1,m+1}$ by $\\hat{u}^{n+1,m}$) become negligible, but the paper does not report iteration counts, tolerances, or a study of how conservation depends on them.","fun_headline_variants_meta":{"raw":{"variants":["Simultaneous relaxation yields exact energy conservation","Low-Mach solver conserves energy to machine precision","Fully conservative scheme for low-Mach flows","All variables relaxed together for exact energy","Energy conservation to roundoff without low-Mach tricks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000437,"raw_usage":{"total_tokens":2266,"prompt_tokens":1031,"completion_tokens":1235,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":647,"completion_tokens_details":{"reasoning_tokens":1165}},"tokens_in":647,"tokens_out":1235,"duration_ms":9006,"temperature":1.0,"reasoning_tokens":1165,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T10:22:33.392184+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Rerun the three-dimensional periodic inviscid test of Section 4.2 with the same grids and time steps but with the simultaneous relaxation limited to 1, 2, 5, 10, and 100 iterations, and record $|\\varepsilon_{\\rho E}|$ at $t/(L/U_0)=10$. If roundoff-level conservation appears only beyond some iteration count, or fails to appear at any finite count, then the conservation claim depends on the loop being run to convergence rather than being an intrinsic property of the discretization. A second check is the entropy error, which the paper reports at $10^{-6}$ rather than roundoff; grid-refinement and iteration-count studies of $|\\varepsilon_{\\rho s}|$ would show whether entropy conservation improves or saturates.","supporting_citations":[],"review_version":1}