{"id":"6fb548e2-829b-4d6c-925b-36a878c79ad8","arxiv_id":"2508.02033","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"A third-order WENO compact least-squares finite volume scheme achieves high accuracy and shock robustness on structured curvilinear non-uniform grids.","lead":"This paper introduces a third-order finite volume scheme that combines weighted essentially non-oscillatory reconstruction with compact least-squares fitting on non-uniform curved grids. It targets hyperbolic conservation laws, including Euler equations, aiming for high accuracy in smooth regions and shock resolution in discontinuous regions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Third-order accuracy claim depends on an unverified shock detector that must never contaminate smooth steep regions; no error bound or isolated test supports this, so the central accuracy assertion is load-bearing and unestablished.","rationale":"I agree with the reader that the improved shock detector is the most fragile load-bearing assumption. The abstract's promise of optimal linear weights in smooth regions and dissipation only near discontinuities is only as strong as the detector's decision boundary. On non-uniform grids this distinction is harder, because mesh stretching itself introduces geometric factors that can resemble large solution gradients. This is a genuine technical risk, not merely an absence of proof: the scheme's effective order is not a property of the reconstruction alone; it depends on the detector switching. I examined the garbled full text for a theorem or a numerical counter-case isolating detector behavior, and none is legible. Since the paper is already marked UNVERDICTED, I do not move the verdict. If a clean text were available and the proposed test passed, the concern would be resolved; if it failed, the accuracy claim would need to be weakened to second-order or the detector redesigned.","tokens_in":11738,"tokens_out":8293,"duration_ms":108308,"concrete_test":"From a clean copy of the paper, run the scheme with and without the shock detector on a smooth manufactured solution containing a resolved steep Gaussian bump (width approximately 8-10 cells at the finest grid) on a non-uniform grid with geometric stretching ratio about 1.2, refining from 64 to 512 cells per dimension. If the detector-enabled run shows L1 or L-infinity convergence order below about 2.8 on the finest two grids, or differs from the detector-disabled run by more than a small tolerance, the detector is contaminating smooth regions and the third-order accuracy claim is empirically false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract asserts that adaptive dissipation, driven by an improved shock detector developed in this work, yields both high resolution in smooth regions and high robustness near discontinuities. For the central claim to hold on non-uniform grids, the detector must classify resolved but steep smooth profiles as smooth; otherwise it activates extra dissipation, the nonlinear weights depart from the optimal linear weights, and the scheme loses the promised third-order accuracy. Conversely, a false negative near a discontinuity would reintroduce oscillations and undermine the essential non-oscillatory property. The supplied full text is unreadable, but the abstract itself provides no detector definition, no threshold or parameter setting, no sensitivity analysis, and no error bound. This is the weakest load-bearing point because it sits between the two halves of the claim: formal accuracy in smooth regions and robustness at shocks. The compact least-squares reconstruction may be third-order in isolation, but the weighted scheme's effective order is controlled by a binary or continuous detector decision; that switching mechanism is the least secure part of the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a third-order weighted essentially non-oscillatory compact least-squares finite volume scheme for hyperbolic conservation laws on structured curvilinear non-uniform grids. The abstract describes compact least-squares reconstruction with optimal linear weights, nonlinear WENO weights, an improved shock detector that adapts dissipation, and extension to Euler equations via characteristic decomposition; numerical examples for linear convection and Euler/Navier-Stokes equations are claimed. The supplied full text, however, is almost entirely unreadable because of encoding corruption, so the derivation, the definition of the shock detector, and the numerical evidence cannot be checked.","tokens_in":11976,"tokens_out":4597,"duration_ms":52286,"significance":"If the claims are correct, the scheme would be a useful addition to high-resolution finite volume methods: it combines third-order reconstruction on non-uniform grids with WENO limiting and a problem-adaptive shock detector, and it addresses the Euler system through characteristic decomposition. The claim is falsifiable and the proposed numerical experiments are standard. However, because no equations, convergence tables, or error statistics can be read in the supplied text, the significance cannot be confirmed at this stage. The paper would benefit from a complete readable manuscript and, ideally, from archived code or data for reproducibility.","major_comments":[{"comment":"The full text supplied after the Abstract is unreadable: it consists of corrupted characters with only isolated readable fragments, and no numbered equations or tables can be audited. This is a load-bearing problem because the central claims of third-order accuracy, essentially non-oscillatory behavior, and robustness are supported only by the abstract. Please provide a clean manuscript with all equations, stencil definitions, detector formulas, and complete numerical tables.","section":"Full Text (after Abstract)"},{"comment":"The improved shock detector is introduced in the Abstract but never defined in any readable part of the text. The detector controls adaptive dissipation, so its false-positive and false-negative behavior directly determines whether the optimal linear weights are preserved in smooth regions and whether oscillations are suppressed near discontinuities. The manuscript should state the detector formula, its thresholds and parameters, and include a sensitivity study or at least a test on a smooth but steep profile on a non-uniform grid to show that the claimed third order is retained.","section":"Abstract"},{"comment":"No convergence study is visible: the abstract mentions numerical examples, but no error versus grid-size table or observed-order calculation can be found in the supplied text. For a third-order claim, the authors should include L1 and L-infinity error tables for the linear advection equation on non-uniform grids, together with the observed order, and a resolution test for a shock-free but high-gradient case.","section":"Full Text, numerical section (unreadable)"}],"minor_comments":[{"comment":"The line 'arXiv:2508.02034v1 [cs.CV] 4 Aug 2025' appears inside the full text; this is extraneous and should be removed.","section":"Full Text, header"},{"comment":"The phrase 'broad-spectrum accuracy' is vague; please define it in terms of formal order and stencil width.","section":"Abstract"},{"comment":"Many of the table fragments are illegible; even after decoding, the tables should include clear captions and identify the error norms and grid parameters.","section":"Full Text, tables"},{"comment":"The reference list, if present in the original, is not readable in the supplied text; the final version must contain complete bibliographic entries for the WENO, compact, and least-squares literature.","section":"Full Text, references"}],"recommendation":"major_revision","confidential_remarks":"The corrupted full text makes normal review impossible; I recommend the editor return the manuscript to the authors to provide a clean, readable version before further review. The embedded arXiv identifier from a different paper (2508.02034v1, cs.CV) should be checked for a copy-paste or submission error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take on 2508.02033. I could not read the paper: the full text I have is garbled OCR, so my judgment rests on the abstract and the shape of the argument. Based on that, this is a plausible, incremental contribution to finite-volume WENO schemes, not a breakthrough. What seems genuinely new is combining compact least-squares reconstruction with optimal linear weights and WENO nonlinear weights on structured curvilinear non-uniform grids, plus an improved shock detector that adjusts dissipation levels, and extending the scheme to the Euler equations via characteristic decomposition. That is a reasonable engineering combination, and the numerical examples claimed (linear convection, Euler/Navier-Stokes) are the right tests if they are reported honestly.\n\nWhat the paper does well, on the abstract's evidence: it identifies a real gap (WENO compact least-squares schemes are usually third-order on uniform grids; extending them to non-uniform curvilinear grids without losing order takes work) and it attempts a practical fix. The mention of explicit second-order polynomials per control volume is a concrete algorithmic statement.\n\nSoft spots. The central accuracy claim is load-bearing and under-supported in the abstract. The scheme's formal third-order accuracy requires that in smooth regions the nonlinear WENO weights coincide with the optimal linear weights; anything that disrupts that - especially the improved shock detector misclassifying a steep but smooth profile as a discontinuity - drops the order. The stress-test note is right that the abstract gives no detector definition, no thresholds, no sensitivity analysis, and no error bound. That is a legitimate referee question, but it is not evidence of a flaw; it is a missing justification. There is also the usual WENO concern about hand-tuned parameters, which the abstract cannot rule out. I cannot assess the numerics at all because the tables and equations are unreadable here.\n\nOverall: if the full text is as coherent as the abstract suggests, this deserves a serious referee. The contribution is small but real, and the shock-detector question is exactly the kind of thing peer review should pressure-test. I would not desk-reject. I also would not cite it from the abstract alone; I need to see the reconstruction formulas, the detector, and the convergence tables.\n\nRecommendation: send it to review, with a referee who knows WENO compact schemes well enough to ask whether the detector preserves optimal weights in smooth regions.","headline":"Third-order WENO compact least-squares on non-uniform grids: plausible incremental contribution, but the supplied text is unreadable so I can only judge it by the abstract.","tokens_in":12430,"tokens_out":2209,"would_cite":false,"duration_ms":23681,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M08","35L65","76M12"],"pacs":["47.11.Df","47.40.-x"],"model":"deepseek-v4-flash","headline":"This paper develops a third-order finite-volume WENO scheme that keeps high accuracy on non-uniform structured grids and stays non-oscillatory at shocks.","keywords":["WENO","compact least-squares reconstruction","finite volume method","hyperbolic conservation laws","non-uniform grids","shock detection","Euler equations","essentially non-oscillatory"],"falsifier":"Run the scheme on a smooth, high-frequency sine solution on a stretched or skewed grid and measure the actual convergence rate; if the shock detector flags smooth extrema and the rate drops below third order as the mesh is refined, the central accuracy claim fails. Equivalently, a detector that labels a smooth but steep tanh profile as discontinuous on a non-uniform mesh would falsify the premise.","tokens_in":11577,"feed_emoji":"🌊","tokens_out":4572,"duration_ms":49600,"temperature":0.7,"pith_summary":"The paper is trying to establish that a finite-volume discretization can have third-order accuracy on smoothly varying structured curvilinear grids and stay essentially non-oscillatory across discontinuities without sacrificing either property. The scheme combines compact least-squares reconstruction, whose optimal linear weights deliver broad-spectrum accuracy in smooth regions, with nonlinear weights that switch behavior at shocks. The new ingredient is an improved shock detector that selects adaptive levels of dissipation locally, keeping smooth regions low-dissipation and discontinuous regions stable. Numerical tests on the linear convection equation and on the Euler and Navier-Stokes equations are offered as evidence that the method resolves fine structure while remaining stable. If true, this gives computational fluid dynamics a practical third-order shock-capturing method on the non-uniform meshes that realistic geometries require.","feed_headline":"Third-order shock scheme keeps accuracy on non-uniform grids","feed_subtitle":"Adaptive-dissipation detector preserves sharp shocks and smooth third-order resolution in compressible-flow tests.","key_machinery":"The carrying mechanism is the compact least-squares reconstruction: for each control volume an explicit second-order polynomial is fit by least squares over a compact stencil, producing optimal linear weights that are exact for smooth data. A WENO-style nonlinear weighting of candidate reconstructions supplies the non-oscillatory safeguard at discontinuities. The improved shock detector is the device that decides how much dissipation to apply: in smooth regions it keeps the optimal linear weights, and where it detects a discontinuity it raises dissipation through adaptive coefficients. For the Euler equations, characteristic decomposition splits the system into independent scalar waves so that the reconstruction and the detector act on each characteristic field separately.","core_discovery":"The central claim is that, on structured curvilinear non-uniform grids, a compact least-squares reconstruction using explicit second-order polynomials per control volume yields optimal linear weights that preserve third-order accuracy in smooth regions; when the improved shock detector flags non-smooth data, nonlinear weights with adaptively increased dissipation restore the essentially non-oscillatory property. The same scheme therefore achieves high resolution in smooth regions and stable, oscillation-free behavior near discontinuities. The method is extended to systems through characteristic decomposition, and the numerical examples on linear convection and nonlinear Euler/Navier-Stokes equations are presented as demonstrations of the method's accuracy and stability.","pith_inferences":["A testable extension the authors do not pursue is a rigorous error estimate for the adaptive dissipation mechanism, showing that the detector's decisions never degrade the formal third-order rate on smooth non-uniform meshes.","The same detector concept may transfer to other high-order finite-volume or discontinuous-Galerkin schemes that need to switch between optimal and nonlinear weights for accuracy versus stability.","Because the detector is the load-bearing component, its reliability on highly skewed or rapidly stretched grids is the most likely place for the method to be tested; a systematic study of the detector's parameters on such meshes would be a natural follow-up."],"forward_implications":["On smoothly varying structured curvilinear grids, the scheme should exhibit third-order convergence for smooth solutions, not only on uniform meshes.","The adaptive dissipation selected by the shock detector should keep discontinuities sharp without spurious oscillations while preserving accuracy away from shocks.","The method applies to scalar conservation laws and to systems such as the Euler equations via characteristic decomposition, making it usable for compressible flow simulations.","The compact stencil and per-control-volume polynomial reconstruction keep the scheme local, which is favorable for parallel computing and for complex geometries.","If the numerical evidence holds, the scheme offers a single finite-volume formulation that handles both smooth flow features and shock waves on non-uniform grids."],"supporting_citations":[],"fun_headline_variants":["Compact least-squares scheme tames shocks on non-uniform grids","Adaptive-dissipation detector keeps shocks crisp on uneven meshes","Shock-capturing third-order scheme for uneven grids","Third-order scheme stays robust on non-uniform grids","Third-order WENO scheme holds accuracy on non-uniform grids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the improved shock detector reliably separates smooth regions from discontinuities on non-uniform grids: if it mistakes a smooth but steep region for a shock, the scheme falls back to extra dissipation and loses the third-order accuracy that the rest of the construction supplies.","fun_headline_variants_meta":{"raw":{"variants":["Compact least-squares scheme tames shocks on non-uniform grids","Adaptive-dissipation detector keeps shocks crisp on uneven meshes","Shock-capturing third-order scheme for uneven grids","Third-order scheme stays robust on non-uniform grids","Third-order WENO scheme holds accuracy on non-uniform grids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001156,"raw_usage":{"total_tokens":4721,"prompt_tokens":812,"completion_tokens":3909,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":3826}},"tokens_in":428,"tokens_out":3909,"duration_ms":27810,"temperature":1.0,"reasoning_tokens":3826,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:12:46.158325+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the scheme on a smooth, high-frequency sine solution on a stretched or skewed grid and measure the actual convergence rate; if the shock detector flags smooth extrema and the rate drops below third order as the mesh is refined, the central accuracy claim fails. Equivalently, a detector that labels a smooth but steep tanh profile as discontinuous on a non-uniform mesh would falsify the premise.","supporting_citations":[],"review_version":1}