{"id":"ebe6030c-4a4b-4ded-a73f-53f4258d511a","arxiv_id":"2502.10289","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A comparison of Euler, Heun, midpoint, RK4, and ODE45 on three case studies claims model-dependent accuracy, but the evidence is inconsistent and not reproducible.","lead":"Five standard numerical ODE solvers are compared on three textbook models: logistic population growth, building temperature change, and market equilibrium price. The paper reports that accuracy depends on the model, but the results are internally contradictory and the experimental setup is not described.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Internal contradictions between the prose and Tables 1–2 invalidate the solver rankings on which the central claim depends.","rationale":"I checked whether the central claim could be supported even under the most charitable reading of the paper's reported numbers. It cannot, because the prose and the tables disagree about which solver has the lowest error. This is not a question of novel methodology or of disagreement with community consensus; it is an internal inconsistency in the evidence. The reader's weakest assumption focused on the unstated numerical implementations and the validity of the experimental data. While those are genuine problems, the more immediate and falsifiable defect is that even the paper's own displayed results do not support the rankings asserted in the text. For example, Table 1 shows Euler with the smallest experimental error, yet §3.1 names Heun and then ode45 as the most accurate; Table 2 shows Midpoint as the smallest, yet §3.2 names ode45 as the most accurate. This makes it impossible to extract a consistent statement of which solver is 'best' for each case study, which is precisely the conclusion the paper claims to establish. I am not alleging any misconduct; the contradictions could arise from typographical errors, mismatched labels, or careless drafting. But the consequence is the same: the central comparative claim is not supported by the evidence as written. The recommendation to reject the paper is therefore unchanged, and this specific internal contradiction should be one of the primary reasons stated in the review.","tokens_in":9126,"tokens_out":2617,"duration_ms":25147,"concrete_test":"Recompute the rank of each solver from the experimental-error columns of Tables 1 and 2 by taking the argmin per table, and compare that argmin with the 'best solver' named in the prose and conclusion. For Table 1, the argmin is Euler (2.9281), not Heun or ode45; for Table 2, the argmin is Midpoint (0.5362), not ode45. If the prose's claimed best solver differs from the table's argmin, the paper's own data contradict its conclusion. To go further, one would need the actual ODEs, initial conditions, step sizes, and reference data used to regenerate the tables; absent those, no external check is possible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of the paper is that the relative accuracy of the five ODE solvers changes with the qualitative behavior of the model, so no single solver is universally best. That claim rests entirely on the error values in Tables 1–3. However, the prose describing those tables contradicts the tables themselves. In §3.1, the text states that Heun's method has the lowest experimental error, and then states that ode45 has the lowest error at 2.9417. Table 1 reports experimental errors of Euler 2.9281, Heun 2.9438, Midpoint 2.9418, RK 2.9417, and ode45 2.9417; Euler is actually the smallest, while Heun is the largest. In §3.2, the text first concludes that Midpoint is best based on experimental error, then states that ode45 'proved to be the most accurate' with the lowest error at 0.6790. Table 2 reports experimental errors of Euler 0.8122, Heun 0.5367, Midpoint 0.5362, RK 0.6182, and ode45 0.6790; Midpoint is the minimum and ode45 is not the lowest. Because the named 'best' solver in the prose does not match the minimum of the corresponding table, the reported rankings are not internally consistent. Furthermore, no ODE equations, step sizes, parameters, or source datasets are provided, so it is impossible to verify any rank order from the paper's own materials. The empirical errors of exactly 100.00 for all five solvers in Table 3 further suggest that the error computation for that case is degenerate or mis-specified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript compares five explicit ODE solvers (Euler, Heun, midpoint, classical RK4, and MATLAB's ODE45) on three case studies: logistic population growth, building temperature change, and market equilibrium price. It reports relative errors computed against 'empirical' and 'experimental' reference data in Tables 1-3 and concludes that no single solver is universally best, that all solvers are nearly equivalent for the population case, that the midpoint method is best for the temperature case, and that none can track the market-price dynamics. The paper provides textbook summaries of the methods but does not state the ODEs, parameters, initial conditions, step sizes, tolerances, or source datasets behind any of the computed errors.","tokens_in":9481,"tokens_out":6858,"duration_ms":63917,"significance":"If the results were reproducible and internally consistent, the paper would provide a modest empirical illustration of a well-known principle: solver choice should depend on problem structure. That principle is already standard in numerical analysis, and the paper offers no new algorithm, theory, code, or data set. The main potential value would be as a reproducible benchmark, but that value is not realized because the experimental setup is almost entirely unspecified. On the positive side, the idea of comparing numerical solutions against both an empirical formula and previously published experimental data is appropriate for the stated applied goal, and the method summaries indicate familiarity with the classical fixed-step schemes. However, the evidence in the tables is not reliable enough to support even the qualitative ranking claims.","major_comments":[{"comment":"The prose and the tables contradict each other on the very rankings on which the paper's conclusions depend. In §3.1 the text first says Heun's method has the lowest experimental error and then says ode45 has the lowest error at 2.9417, but Table 1 gives Euler 2.9281, Heun 2.9438, midpoint 2.9418, RK4 2.9417, and ode45 2.9417; Euler is the minimum and Heun is the maximum. In §3.2 the opening paragraph concludes the midpoint method has the lowest experimental error, but the next paragraph says ode45 'proved to be the most accurate' with the lowest error at 0.6790; Table 2 gives midpoint 0.5362 and ode45 0.6790, so the minimum is midpoint and ode45 is not the lowest. These contradictions make it impossible to know which ranking, if any, is the intended result and invalidate the central claim that accuracy ordering changes with model behavior.","section":"§3.1, Table 1; §3.2, Table 2; §4"},{"comment":"The manuscript does not provide the ODEs being solved, their parameter values, initial conditions, integration intervals, step sizes, solver tolerances, or the 'experimental data of previous literature' used as references. Without this information, the relative errors in Tables 1-3 and the rankings drawn from them cannot be reproduced or independently checked. Equations (23) and (24) cannot be evaluated either, because the sets E_i, P_i, and X_i are never defined in concrete terms. This is not a presentation issue: every numerical claim in the paper rests on these unreported inputs.","section":"§2, §3 (all case studies)"},{"comment":"The 'empirical error' column of Table 3 is 100.00 for all five solvers, yet the text interprets the methods as failing in different degrees. The fact that all values are exactly the same strongly suggests that the error computation is degenerate or mis-specified for this case. The explanation in §3.3 that the steep slope 'makes the denominator of all the solvers to be a very small number' is not meaningful without the actual formula and denominators; a blow-up of the numerical solution should not automatically produce an identical error for every method. The authors need to report how Eq. (23) behaves for this case study and why the comparison is not vacuous.","section":"§3.3, Table 3"},{"comment":"The description of ODE45 as a 'variable-step, variable-order algorithm' is not accurate: MATLAB's ode45 is an adaptive step-size implementation of an explicit Runge-Kutta (4,5) pair (Dormand-Prince), not a variable-order method. More importantly, the comparison of fixed-step solvers with an adaptive solver is only meaningful if step sizes and tolerances are chosen under a stated protocol; since no protocol is given, the 'solution cost' claims in §3.1 and the accuracy ranking in §3.2 are not well-defined.","section":"§2.6, §3.2"}],"minor_comments":[{"comment":"Keywords contain 'Euler'r method' and the introduction uses 'Hyun's method'; these should be 'Euler's method' and 'Heun's method'.","section":"Keywords; §1"},{"comment":"Equation (1) and the surrounding text contain corrupted symbols (e.g., 'd!d\"' instead of dy/dx), and Eqs. (23)-(24) do not define n, E_i, P_i, or X_i in a way that makes the computed quantities unambiguous. Please use standard notation and state whether the reported errors are percentages or normalized RMSE values.","section":"§2, Eqs. (1), (23), (24)"},{"comment":"In the sentence after Figure 2, 'form 2' should be 'figure 2'; the same typo appears in §3.1 and §3.2 for 'form' versus 'from'. Additionally, §3.1 refers to the 'exponential model' while the abstract and introduction describe the first case study as a logistic population model; the discrepancy should be resolved.","section":"§3.2"},{"comment":"The introduction states that Heun's method 'enhances accuracy through dynamic step size adaptation,' but the method described in §2.2 is a fixed-step predictor-corrector; this should be corrected.","section":"§1"}],"recommendation":"reject","confidential_remarks":"The paper reads as an unreviewed preprint rather than a finished journal submission. The reference list includes numerous citations to the authors' own works, several with no evident connection to the topic, which may inflate the bibliography. The main issue is not a disagreement with the authors' qualitative thesis but the unreproducible and internally contradictory evidence offered for it. I would not send this back for a standard revision; the authors would need to restate all models, provide data and solver settings, and reconcile the text with the tables before the claims can be evaluated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the paper is a straightforward application of five textbook ODE solvers to three textbook models. There is no new mathematics, no code, no data, and no reproducible workflow. Second, and more seriously, the prose and the tables contradict each other on which solver is best in two of the three case studies, so the central ranking claims are not internally consistent.\n\nWhat it does well: the methodology section is a clean recap of Euler, Heun, midpoint, RK4, and ode45, and the paper cites the relevant prior benchmarking literature (refs 23 and 25). It also makes the sensible observation that solver performance can depend on the model's behavior, and it correctly reports that all five solvers blow up on the market-price model.\n\nThe soft spots are load-bearing. In §3.1, the text first says Heun has the lowest experimental error, then says ode45 has the lowest error at 2.9417. Table 1 shows Euler at 2.9281 (lowest) and Heun at 2.9438 (highest). In §3.2, the text first concludes Midpoint is best, then says ode45 is most accurate at 0.6790. Table 2 shows Midpoint at 0.5362 (lowest) and ode45 at 0.6790. The conclusion repeats the Midpoint claim. These are not typos; they invalidate the reported rankings. Additionally, the paper gives none of the ODE equations, initial conditions, step sizes, or source datasets, so the error values in Equations (23)–(24) cannot be verified. The empirical errors of exactly 100.00 for all five solvers in Table 3 suggest the error computation for that case is degenerate or mis-specified.\n\nThis paper is for a reader who wants a very basic demonstration that solver choice matters. That message is true, but it is not new, and the internal contradictions make it unreliable as a benchmark. It does not deserve a serious referee. Recommend desk rejection; if the authors resubmit, they need to provide the full problem definitions and align the text with the tables.","headline":"A basic ODE-solver benchmark whose reported rankings are internally inconsistent and unreproducible; the paper's central claim doesn't survive scrutiny.","tokens_in":9928,"tokens_out":2521,"would_cite":false,"duration_ms":22276,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65L05","65L06"],"pacs":[],"model":"deepseek-v4-flash","headline":"No single ODE solver is universally best; the model's behavior determines which solver is most accurate.","keywords":["numerical ODE solvers","Euler method","Heun method","midpoint method","Runge-Kutta 4th order","ODE45","relative error","case-study comparison"],"falsifier":"A decisive test would be to re-run the five solvers on the three case studies with the ODEs, step sizes, and reference data supplied, and check whether the midpoint method still has the smallest experimental relative error on the building-temperature case and whether all five solvers still blow up on the market-price case. If, under identical conditions, a different solver wins the temperature case or any solver tracks the market price, the paper's core claim of model-dependent rankings would need revision.","tokens_in":8954,"feed_emoji":"📊","tokens_out":8069,"duration_ms":69200,"temperature":0.7,"pith_summary":"This paper sets out to show that the relative accuracy of five standard explicit ODE solvers — Euler, Heun, midpoint, Runge–Kutta 4th order, and MATLAB's ode45 — is not fixed but changes with the qualitative behavior of the model being integrated. Comparing each solver against empirical and experimental reference data on three case studies, the paper finds that for smooth logistic population growth all five produce nearly identical results, so computational cost decides which is best; for building-temperature dynamics the midpoint method yields the smallest relative experimental error even though it is not the highest-order method; and for market-equilibrium price, which changes suddenly and steeply, none of the five can track the behavior and all blow up in prediction. The paper's takeaway is that solver choice should be guided by the nature of the model, not by a general preference for higher-order or adaptive methods. If this is right, benchmark conclusions about 'the best solver' from one application should not be assumed to transfer to another.","feed_headline":"Best ODE solver depends on the model's behavior","feed_subtitle":"Accuracy rankings flipped across population, temperature, and price cases; model shape matters.","key_machinery":"The central object is the normalized relative-error measure defined in Eqs. (23) and (24), which compares each solver's output to an empirical-model reference and to experimental reference values. The five solvers themselves form the second ingredient: all are explicit one-step methods of the form $y_{i+1}=y_i+\\Phi h$, differing only in how the representative slope is computed. The relative-error measure is what the paper uses to rank the solvers and to conclude that the rankings shift across the three case studies.","core_discovery":"The central claim is that estimation accuracy of numerical ODE solvers is case-dependent. Using the logistic population model, a building-temperature model, and a market-equilibrium-price model, the paper computes relative errors via Eqs. (23) and (24) for each of five solvers against both empirical equation outputs and experimental data from previous literature. The reported result is that the solvers are nearly indistinguishable in relative error on population dynamics; that the midpoint method has the lowest experimental relative error on the temperature case; and that all solvers fail on the market-price case because the model's steep slope makes denominators in the numerical formulas very small. The general conclusion the authors draw is that the application and accuracy of each solver differ according to the behavior of the model.","pith_inferences":["A natural extension the paper does not pursue is to construct a benchmark suite containing several qualitative regimes (smooth, transient, stiff, discontinuous) and report solver rankings per regime; if the rankings still separate cleanly by regime, the model-dependence claim becomes a design principle for solver selection.","Because the paper does not report the underlying ODEs, parameters, step sizes, or the reference datasets, the specific rankings (e.g., midpoint best for temperature) are not yet reproducible; publishing those details would turn the claim into a testable benchmark.","The market-equilibrium failure hints that an implicit or special stiff solver, or an event-detecting formulation, could handle the steep-slope behavior; testing that is a direct, concrete follow-up.","The internal inconsistency between the table (midpoint lowest temperature error) and the prose (ode45 named most accurate) suggests the reported 'best solver' label should be read with care until the dataset and error computations are confirmed."],"forward_implications":["For smooth, logistic-type growth, choosing the solver with the lowest computational cost (Euler's method) loses no meaningful accuracy, so expensive higher-order solvers are unnecessary there.","Higher-order and adaptive solvers are not automatically more accurate: on the building-temperature case the midpoint method is reported to have the smallest experimental relative error despite being only second-order.","For models with sudden, steep changes (market equilibrium price), standard explicit solvers diverge and blow up, so such models require different numerical treatment or model reformulation before these solvers can be used.","A solver ranking established on one case study should not be assumed to carry over to another case study with different qualitative dynamics.","The relative-error formulas in the paper provide a portable, quantitative way to compare solver accuracy against empirical and experimental references in future case studies."],"supporting_citations":[{"why":"It provides the broader benchmarking context of numerical ODE solvers on biological system models, the kind of comparison this paper extends to three new case studies.","marker":"[23]"},{"why":"It supplies a prior comprehensive comparison of ODE solvers that motivates the paper's comparative error-analysis approach.","marker":"[25]"},{"why":"It gives the poultry production model underlying the logistic population-dynamics case study.","marker":"[26]"},{"why":"It provides experimental thermal-comfort data from educational buildings that the temperature case study draws on as reference.","marker":"[28]"},{"why":"It supplies the conceptual dynamic-modeling background for evolving systems used in the market-equilibrium-price case study.","marker":"[33]"},{"why":"It is the source of the numerical method descriptions for Euler, Heun, and midpoint methods applied in the paper.","marker":"[34]"},{"why":"It supports the midpoint method's connection to Newton–Cotes quadrature, which justifies treating it as a distinct solver.","marker":"[36]"}],"fun_headline_variants":["No universal best ODE solver across cases","Solver accuracy flips with model behavior","Midpoint wins temperature; no solver handles price","All solvers tie on population, fail on price","Best ODE solver? It depends on the model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rankings stand on the assumption that the empirical and experimental reference data are valid, aligned at matching time points, and correctly transcribed into the error formulas of Eqs. (23) and (24), and that each solver was implemented and configured fairly, since the paper does not report the ODEs, parameters, step sizes, or source datasets.","fun_headline_variants_meta":{"raw":{"variants":["No universal best ODE solver across cases","Solver accuracy flips with model behavior","Midpoint wins temperature; no solver handles price","All solvers tie on population, fail on price","Best ODE solver? It depends on the model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1559,"prompt_tokens":914,"completion_tokens":645,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":574}},"tokens_in":530,"tokens_out":645,"duration_ms":6418,"temperature":1.0,"reasoning_tokens":574,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T18:38:56.450794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive test would be to re-run the five solvers on the three case studies with the ODEs, step sizes, and reference data supplied, and check whether the midpoint method still has the smallest experimental relative error on the building-temperature case and whether all five solvers still blow up on the market-price case. If, under identical conditions, a different solver wins the temperature case or any solver tracks the market price, the paper's core claim of model-dependent rankings would need revision.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the broader benchmarking context of numerical ODE solvers on biological system models, the kind of comparison this paper extends to three new case studies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies a prior comprehensive comparison of ODE solvers that motivates the paper's comparative error-analysis approach."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the poultry production model underlying the logistic population-dynamics case study."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides experimental thermal-comfort data from educational buildings that the temperature case study draws on as reference."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the conceptual dynamic-modeling background for evolving systems used in the market-equilibrium-price case study."},{"cited_title":"V., & Sherman, A","cited_arxiv_id":null,"evidence_quote":"It is the source of the numerical method descriptions for Euler, Heun, and midpoint methods applied in the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supports the midpoint method's connection to Newton–Cotes quadrature, which justifies treating it as a distinct solver."}],"review_version":1}