{"id":"2a5f413e-3781-4980-94a6-c19b14c99567","arxiv_id":"2601.17452","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Different high-order numerical schemes for the compressible Euler equations yield different scheme-dependent dissipative weak solutions in oscillatory regimes, while order-averaged and time-averaged statistics look more consistent.","lead":"This numerical study runs first- through ninth-order shock-capturing schemes on 2-D Riemann problems and a shear-layer instability test, finding that different schemes do not settle on the same limiting solution when the flow becomes oscillatory. The authors interpret these scheme-dependent limits as dissipative weak solutions of the Euler equations and compare them with order-averaging, Young-measure histograms, entropy production, and energy defects.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central DW-limit claim depends on equating order-averages at one fixed mesh with K-convergence; no consistency proof or mesh refinement supports this, so the scheme-dependent limits are not established.","rationale":"The reader identified the core interpretive gap: Theorem 2.5 applies to consistent approximations on refined meshes, whereas the paper averages over formal order at a fixed 1024×1024 mesh, and no consistency analysis is available for the high-order schemes. My stress-test agrees and finds this to be the single most load-bearing concern, because the entire conclusion that the limits are DW solutions depends on this identification. The paper's own admission in Section 1 and the lack of mesh-refinement Cesàro averages make the claim an exploratory numerical indication rather than a demonstrated instance of K-convergence. I considered other potential objections—such as the spatial-histogram approximation of Young measures and the use of only six orders—but these are subsidiary to the missing consistency/mesh-refinement link. The proposed concrete test directly checks both conditions of Theorem 2.5: strong convergence of Cesàro averages along refined meshes, and vanishing consistency residuals. If those pass, the central claim is supported; if they fail, the paper should be reframed as a numerical study of fixed-resolution behavior rather than of DW-limit convergence. Since the reader's verdict is already CONDITIONAL and my analysis confirms it, no change to the verdict is needed.","tokens_in":20917,"tokens_out":2891,"duration_ms":30918,"concrete_test":"Perform a genuine K-convergence check for the KH instability (or Configuration 3): for one fixed scheme order (e.g., fifth-order LCDCU, LDCU, and VFV), compute solutions on refined meshes h=1/256, 1/512, 1/1024, 1/2048. Form Cesàro averages ρ̄_N = (1/N)Σ_{n=1}^N ρ_{h_n} and test (i) whether ||ρ̄_N − ρ̄_M||_{L1(Ω)} → 0 as N,M → ∞, and (ii) whether the weak-form residuals of the continuity and momentum equations with smooth compactly supported test functions tend to 0 as h→0, i.e., whether consistency in the sense of Definition 2.1 holds. If the mesh-refined Cesàro averages converge strongly and the limits differ across schemes, the paper's central claim is corroborated; if the residuals do not vanish or the averages do not stabilize, the order-averaged interpretation at fixed mesh is not a valid instance of K-convergence and the claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main conclusion—that different numerical methods converge to different DW solutions—rests on identifying order-averaged quantities (eρ_n, etc.) with the K-convergence Cesàro averages of Theorem 2.5. That theorem requires a consistent approximation indexed by mesh refinement h_n→0 and then averaging subsequences as n→∞. Here the averaging index is formal order of accuracy (ℓ=1,...,6) at a single mesh Δx=Δy=1/1024. The paper explicitly states in Section 1 that 'a rigorous consistency analysis is not available for high-order methods,' so the high-order sequences are not known to satisfy Definition 2.1; the consistency errors e_ℓ[φ] are never measured. Moreover, the 'convergence of averages over orders' (Section 4.3) is assessed only for n=1,...,5 against n=6 as reference, with fitted power laws (Table 4.2) but no evidence that the n→∞ limit exists or that it is independent of the finite set of orders. The Young-measure PDFs are also spatial histograms over a subdomain, not pointwise Young measures, and their scheme dependence is taken as evidence of distinct DW limits. Without a valid K-convergence instance—mesh refinement, verified consistency, and Cesàro averages along a subsequence—the central claim that order-averaged solutions converge strongly to scheme-dependent DW solutions is unsupported, however plausible the numerics may be.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports a numerical study of the compressible Euler equations using three families of high-order schemes (LCDCU, LDCU, VFV) at formal orders 1, 2, 3, 5, 7, and 9, applied to two-dimensional Riemann problems and a Kelvin-Helmholtz instability. The central interpretive claim is that, in oscillatory regimes, different numerical methods converge only weakly and that averages over the different formal orders at a fixed grid converge strongly, in the sense of K-convergence, to scheme-dependent dissipative weak (DW) solutions. The paper also computes approximate Young-measure PDFs and compares entropy production and energy defects as potential selection criteria.","tokens_in":21312,"tokens_out":5026,"duration_ms":54611,"significance":"If the central claim were justified, the paper would provide valuable numerical evidence for the non-uniqueness of DW solutions and for scheme-dependence of limits of oscillatory numerical approximations, with practical criteria for selecting physically relevant solutions. The manuscript also makes a useful contribution by extending a family of A-WENO schemes up to ninth order and by comparing three different numerical fluxes on challenging benchmark problems. The authors are transparent about the lack of a rigorous consistency analysis for high-order methods and make the code available on request. However, the main theoretical interpretation is not supported by the experiments as presented.","major_comments":[{"comment":"The averaging procedure used in the numerical study is not an instance of the K-convergence theorem. Theorem 2.5 applies to a consistent approximation indexed by mesh refinement h_n → 0 and requires Cesàro averages along a subsequence. The paper instead averages six solutions of different formal orders at the single mesh Δx = Δy = 1/1024. Moreover, Section 1 states that 'a rigorous consistency analysis is not available for high-order methods', so the sequences are not known to satisfy Definition 2.1, and the consistency errors e[φ] are never measured. The abstract and Section 5 claim that order-averaged solutions converge to DW solutions in the K-convergence sense; this is unsupported.","section":"Section 4 (paragraph after notation) and Theorem 2.5"},{"comment":"The evidence for 'K-convergence of the sequence of averages' consists of the L1 errors ||ρ̄_n - ρ̄_6|| for n = 1,...,5 relative to the sixth-order reference. This measures only closeness to a particular finite-order average; it does not establish existence of a limit as n → ∞. With n ≤ 6 and no mesh refinement, the fitted power laws in Table 4.2 are descriptive curve fits, not convergence tests. The claim that the average sequence 'is convergent' is therefore not demonstrated.","section":"Section 4.3, Figure 4.13, Table 4.2"},{"comment":"The approximate PDFs σ_ℓ and σ̄_n are spatial histograms of density values over a subdomain Ω̃, not the parametrized Young measure V_{x,y,t} appearing in Definition 2.3. The latter is a pointwise measure in phase space at each (x,y,t). A spatial histogram combines oscillations in space and time and cannot, by itself, establish that the limiting DW solutions differ. The connection between the computed histograms and the measure-valued limit is not made rigorous.","section":"Sections 4.1–4.2, Figures 4.7 and 4.12"},{"comment":"The selection criteria are evaluated on S_n(t) and D_E^n(t) computed from order-averaged fields. Since these fields are not shown to be K-convergent limits, the comparisons may reflect properties of the averaging procedure rather than of genuine DW solutions. In addition, S_n(t) is the total entropy, not an entropy production rate; the paper should define the 'entropy production' that criteria (i) refers to, and clarify how it is measured.","section":"Section 4.4, Tables 4.3 and 4.4"}],"minor_comments":[{"comment":"The continuity equation consistency error e_2[φ] is introduced for φ ∈ C^1_c, but the convergence condition is stated for φ ∈ C^2_c. Please clarify the required test-function regularity.","section":"Definition 2.1"},{"comment":"The averaged quantities ρ̄_n, m̄_n, S̄_n are introduced informally. Please give explicit definitions, e.g., ρ̄_n = (1/n) Σ_{ℓ=1}^n ρ_ℓ, and state that these are functions on Ω×(0,T).","section":"Section 4 notation"},{"comment":"The least-squares fits use only five data points (n = 1,...,5). Report the number of degrees of freedom or confidence intervals so the reader can judge the fit quality.","section":"Table 4.2"},{"comment":"The Heaviside functions in the A-WENO flux formulas make the order selection implicit. A short table or sentence summarizing which correction terms are active for r = 3,5,7,9 would improve readability.","section":"Section 3.2"},{"comment":"The data availability statement says FORTRAN codes are available upon request. A permanent repository or DOI, and a description of the pseudo-random number generation for the KH initial data, would strengthen reproducibility.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"The numerical experiments are extensive and the authors are well known in the field, but the central claim—that the order-averages converge to DW solutions in the K-convergence sense—is not supported by the current computations. The gap is not a minor omission: Theorem 2.5 requires consistency on refined meshes, whereas the paper averages over formal orders at one fixed resolution. I recommend major revision: either add a mesh-refinement study that verifies consistency and demonstrates actual K-convergence for at least one of the schemes, or substantially weaken the claims to a heuristic/numerical-observation level. The selection criteria section also needs a clearer connection to the theoretical criteria in the cited preprints, which are not yet published."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a useful numerical study, and the authors are upfront about the gaps in the theory. The strongest claim, that different schemes converge to different dissipative weak solutions, is presented as a demonstration, but it isn't supported by the cited K-convergence theorem as applied here.\n\nWhat's new and good: the paper gives a systematic comparison of first- through ninth-order LCDCU, LDCU, and VFV schemes on several 2-D Riemann problems and a KH instability. The order-averaged density fields, the normalized histograms meant to approximate Young-measure PDFs, and the entropy/energy-defect tables are concrete and useful. The scheme implementations are spelled out to the point where the experiments are reproducible, and the random data for the KH initial condition are tabulated. The prose is honest: Section 1 admits no rigorous consistency analysis exists for high-order methods, and Section 4.2 says the results 'indicate' scheme-dependent limits rather than claiming proof.\n\nThe soft spot: the paper identifies its order-averages on a fixed 1024×1024 grid with the Cesàro averages of Theorem 2.5. That theorem requires a consistent approximation indexed by mesh refinement, h_n→0. Averaging over formal orders at one mesh is not an instance of it. Section 4.3 says the decay of ||eρ_n−eρ_6||_L1 'demonstrates K-convergence,' but decay toward a particular reference solution of the same grid isn't convergence to a DW limit. Also, the claim that different orders converging weakly 'demonstrates that the consistency errors decay when the mesh is refined' is a non-sequitur: order refinement at fixed h says little about h→0 consistency. The PDFs are spatial histograms over subdomains, which can be scheme-dependent without pinning down distinct DW solutions. These are real problems for the central interpretation, and they aren't manufactured.\n\nProportion: the experiments themselves stand. The authors only need to either add mesh-refinement Cesàro averages (say, first and third order on a few refined grids) or soften the conclusion to what the numerics actually support. Releasing the FORTRAN code would also help.\n\nWho this is for: people designing or validating schemes for under-resolved compressible flows, and anyone working on measure-valued / dissipative solution frameworks. I'd bring it to a reading group to argue about exactly this gap.\n\nRecommendation: send it to peer review. A good referee will ask for the refinement study or a rewrite of the claim. The numerical core is worth publishing.","headline":"A careful numerical comparison whose central interpretive claim—order-averages at one grid being K-convergence—doesn't follow from the cited theorems.","tokens_in":21781,"tokens_out":2675,"would_cite":false,"duration_ms":29337,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M08","65M12","76M12","76M20","76N10","35L65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Different numerical schemes converge to different dissipative weak solutions of the compressible Euler equations.","keywords":["dissipative weak solutions","Euler equations","gas dynamics","K-convergence","Young measures","measure-valued solutions","Kelvin-Helmholtz instability","energy defect"],"falsifier":"Run one of the schemes on the Kelvin-Helmholtz test at meshes 256, 512, 1024, and 2048 and form order-averaged densities at each resolution. If these averages do not converge, or converge to different limits on different refinement paths, the claim that order-averaging yields strong convergence to a single DW solution fails. If a very-high-resolution reference simulation shows all schemes' averages approaching one common field, then the scheme-dependence of the DW limit would be contradicted.","tokens_in":20803,"feed_emoji":"🌀","tokens_out":5673,"duration_ms":60183,"temperature":0.7,"pith_summary":"The paper studies first- through ninth-order versions of three finite-volume and finite-difference schemes applied to two-dimensional Riemann problems and a Kelvin-Helmholtz instability. It argues that when the flow develops fine-scale oscillations, the numerical approximations do not converge strongly to a single weak solution; instead, their weak limits are generalized dissipative weak solutions that depend on the numerical scheme. The paper shows that averages of solutions across orders of accuracy converge strongly, and time-averaged flows look similar across methods, even when pointwise fields differ. It proposes entropy production and energy defect as criteria to select among these non-unique limits. A sympathetic reader would care because this pins down what simulations of turbulent compressible flow actually compute.","feed_headline":"Numerical schemes pick different weak limits for turbulent gas flows","feed_subtitle":"Averaging across solver orders stabilizes the answer, but pointwise fields still depend on the scheme.","key_machinery":"The load-bearing object is the dissipative weak solution: a measure-valued solution in which the momentum equation carries an additional Reynolds stress defect and the total energy carries an energy defect, both encoded in a Young measure over density, momentum, and entropy. The mechanism used to extract a strong limit is K-convergence: averaging a weakly convergent sequence of approximations (here, averages over the six formal orders at fixed mesh resolution) can produce strong convergence to a DW solution. The paper also uses the time-average of the averaged density and histogram approximations of the Young measure's probability density to assess convergence and scheme dependence.","core_discovery":"The paper's central claim is that in regimes where the Euler equations develop fine-scale oscillations, the limit of a consistent numerical approximation is not a unique weak solution but a generalized dissipative weak solution that depends on the chosen numerical method. Using first- through ninth-order versions of three finite-volume/finite-difference methods on the same problems, the authors observe that pointwise density fields from different schemes diverge, while order-averaged densities converge within each scheme yet differ across schemes. They interpret the scheme-dependent limits as different DW solutions, characterized by non-Dirac Young measures, and use entropy-production and en","pith_inferences":["The identification of order-averaging on one fixed grid with the K-convergence averaging of the theory is an extrapolation: the theorem requires a sequence of consistent approximations on refined meshes, while the experiments average six orders on one mesh and the paper concedes that rigorous consistency for the high-order schemes is not available.","If scheme-dependent DW limits occur generically, then a practical consequence is that uncertainty in turbulent compressible-flow simulations should be quantified not just by mesh-refinement error but by comparing different numerical fluxes; convergence of a single scheme may be convergence to a scheme's own DW limit.","The robustness of time-averaged order-averaged fields suggests a testable extension: statistical quantities such as mean density, momentum, or energy across an ensemble of initial perturbations should be reproducible across qualitatively different numerical methods even when snapshot fields are not.","A natural next step the paper does not take is to prove uniqueness of the DW solution that maximizes entropy production (or extremizes energy defect) within a suitable class; if such uniqueness held, the selection criteria would promote an observation about schemes into a definition of the physically relevant limit."],"forward_implications":["In oscillatory flow regimes, no single consistent numerical method is guaranteed to approximate the same weak solution; the scheme selects a particular dissipative weak limit.","Averaging approximations of increasing formal order at a fixed resolution yields strongly convergent-looking sequences, and these order-averaged fields can distinguish genuine DW solutions from plain weak solutions.","Time-averaged density fields of order-averaged solutions look nearly identical across different schemes, suggesting that statistical measures are more transferable than pointwise fields.","The limiting Young measures are not Dirac deltas in regions with oscillations, and their approximated probability densities depend on the numerical scheme.","Entropy production and energy defects provide computable selection criteria, but the rankings of schemes differ depending on which criterion is used; the paper leaves the choice open."],"fun_headline_variants":["Scheme-dependent weak limits for gas dynamics","Which solver? Which weak solution for turbulent gas?","Turbulent gas: numerical scheme sets the weak limit","Different solvers yield different gas flow limits","Numerics decide the weak solution in gas dynamics"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that averaging six numerical solutions of different formal orders on one fixed 1024 by 1024 mesh behaves like the K-convergence averaging from the theory, which applies to subsequences of consistent approximations on refined meshes converging to zero mesh size. The paper itself states that rigorous consistency is not available for the high-order schemes, so this identification is an assumption rather than a theorem.","fun_headline_variants_meta":{"raw":{"variants":["Scheme-dependent weak limits for gas dynamics","Which solver? Which weak solution for turbulent gas?","Turbulent gas: numerical scheme sets the weak limit","Different solvers yield different gas flow limits","Numerics decide the weak solution in gas dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1227,"prompt_tokens":667,"completion_tokens":560,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":489}},"tokens_in":411,"tokens_out":560,"duration_ms":9814,"temperature":1.0,"reasoning_tokens":489,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T08:17:08.507016+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run one of the schemes on the Kelvin-Helmholtz test at meshes 256, 512, 1024, and 2048 and form order-averaged densities at each resolution. If these averages do not converge, or converge to different limits on different refinement paths, the claim that order-averaging yields strong convergence to a single DW solution fails. If a very-high-resolution reference simulation shows all schemes' averages approaching one common field, then the scheme-dependence of the DW limit would be contradicted.","supporting_citations":[],"review_version":1}