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REVIEW 4 major objections 4 minor 36 references

Investigation of the Estimation Accuracy of 5 Different Numerical ODE Solvers on 3 Case Studies

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read No single ODE solver is universally best; the model's behavior determines which solver is most accurate.

desk verdict A basic ODE-solver benchmark whose reported rankings are internally inconsistent and unreproducible; the paper's central claim doesn't survive scrutiny. read the letter →

arxiv 2502.10289 v2 pith:IRLPAYI5 submitted 2025-02-14 math.NA cs.NAmath.AGmath.DG

Hamidreza Moradi , Hamideh Hossei This is my paper · ORCID
classification math.NAcs.NAmath.AGmath.DG MSC 65L0565L06
keywords numericalODEsolversEulermethodHeunmidpointRunge-Kutta4thorderODE45relativeerrorcase-studycomparison
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

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.

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 (4)
  1. [§3.1, Table 1; §3.2, Table 2; §4] 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.
  2. [§2, §3 (all case studies)] 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.
  3. [§3.3, Table 3] 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.
  4. [§2.6, §3.2] 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.
minor comments (4)
  1. [Keywords; §1] Keywords contain 'Euler'r method' and the introduction uses 'Hyun's method'; these should be 'Euler's method' and 'Heun's method'.
  2. [§2, Eqs. (1), (23), (24)] 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.
  3. [§3.2] 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.
  4. [§1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solver-error rankings are compared against external literature data, not derived from the rankings themselves.

full rationale

The paper's central claim is that the relative accuracy of five standard ODE solvers depends on the qualitative behavior of the modeled system. The error values in Tables 1-3 are computed with Equations (23) and (24), which compare each solver's output X_i against external empirical data P_i and empirical-equation data E_i. No solver parameter is fitted to the reported error values, and the conclusion is not used to define the error metric. The methods themselves are textbook formulas (Equations 3, 12, 14, 16-18, and 22), and no load-bearing result is imported from the authors' own prior work. The prose contradicts Tables 1 and 2 when naming the 'best' solver, and the source ODEs, step sizes, and literature datasets are not specified, making the reported rankings unreproducible; however, these are correctness and reproducibility defects, not circularity. There is no exhibited reduction in which an input is defined in terms of the claimed output, so the honest finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central comparisons depend on unstated implementation choices (step size, tolerance, model parameters) and on the accuracy of external experimental data that are never specified. No invented entities are introduced.

free parameters (3)
  • Step size h = Not stated
    The error values in Tables 1 to 3 depend directly on h; without it, the comparison cannot be reproduced.
  • ODE45 error tolerance = Not stated
    ODE45's adaptive step selection and reported error depend on tolerance settings not given in Section 2.6.
  • Model parameters for the three case studies = Not stated
    Logistic growth rate and capacity, building thermal coefficients, and market supply and demand parameters are never defined in Section 3.
assumptions (3)
  • standard math Taylor series expansion of the solution with remainder, used to derive Euler's truncation error in Equations (5) through (8).
    This is standard analysis and not a weakness, but it is an unproved background result the error discussion relies on.
  • domain assumption The empirical relationship and experimental data from prior literature are accurate, comparable, and correctly used as ground truth.
    Invoked throughout Section 3 to claim 'good accordance' and to compute errors in Tables 1 to 3, but the source data are never identified.
  • ad hoc to paper The numerical solvers were all run with step sizes and settings that make their errors directly comparable.
    No step size or tolerance is reported, yet the relative ordering of errors depends on these choices.

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Cite this review

Pith. "Pith review of Investigation of the Estimation Accuracy of 5 Different Numerical ODE Solvers on 3 Case Studies." pith.science (2026). https://pith.science/paper/IRLPAYI5

@misc{pith2026250210289,
  author       = {Pith},
  title        = {Pith review of: Investigation of the Estimation Accuracy of 5 Different Numerical ODE Solvers on 3 Case Studies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRLPAYI5}},
  note         = {Machine review of arXiv:2502.10289}
}
read the original abstract

Numerical ordinary differential equation (ODE) solvers are indispensable tools in various engineering domains, enabling the simulation and analysis of dynamic systems. In this work, we utilize 5 different numerical ODE solvers namely: Euler's method, Heun's method, Midpoint Method, Runge-kutta 4th order and ODE45 method in order to discover the answer of three wellknown case studies and compare their results by calculation of relative errors. To check for the validity of the estimations, the experimental data of previous literature have been compared with the data in this paper which shows a good accordance. We observe that for each of the case studies based on the behavior of the model, the estimation accuracy of the solvers is different. For the logistic population change as the first case study, the results of all solvers are so close to each other that only their solution cost can be considered for their superiority. For temperature change of a building as the second case study we see that in some especial areas the accuracy of the solvers is different and in general Midpoint ODE solver shows better results. As the last case study, market equilibrium price shows that none of the numerical ODE solvers can estimate its behavior which is due to its sudden changing nature.

Figures

Figures reproduced from arXiv: 2502.10289 by the authors.

Figure 2
Figure 2. Comparison of 5 considered ODE solvers on temperature change [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Comparison of 5 considered ODE solvers on market equilibrium price [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

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Reviewed August 7, 2026 · model on record in the stance chip above.