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REVIEW 1 major objections 7 minor 58 references

Some properties of high-order nonstandard multistep multistage methods

T0 review · 1 major / 7 minor · reviewed 2026-07-10 · glm-5.2

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Pith's one-line read Bounded step-size replacement preserves order and stability

desk verdict Solid extension of nonstandard finite differences to multistep multistage methods; the math is correct and the contribution is real but incremental. read the letter →

arxiv 2607.08694 v1 pith:SOEXC25W submitted 2026-07-09 math.NA cs.NA

Balint M. Takacs This is my paper · ORCID
classification math.NAcs.NA
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The pith

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

The reading

The paper introduces nonstandard multistep multistage methods by taking a known class of explicit time-stepping schemes (written in the Shu-Osher strong-stability-preserving form) and replacing every occurrence of the step size Δt with a bounded function φ(Δt). The central result is a pair of equivalences. First, the nonstandard method converges if and only if the original standard method converges, because consistency and stability are unchanged by the substitution. Second, the nonstandard method attains the same order of accuracy p as the standard method if and only if the Taylor expansion of φ matches the identity function up to p+1 terms, i.e., φ(h) = h + O(h^{p+1}). Third, if φ is bounded above by C·B_FE, where C is the SSP coefficient of the method and B_FE is the step-size limit under which forward Euler preserves the desired qualitative property, then the nonstandard method preserves that property (non-negativity, boundedness, mass conservation) for every positive Δt. The mechanism is that the SSP form decomposes the method into a convex combination of forward Euler steps with effective step sizes (β/α)·φ(Δt), so bounding φ keeps each sub-step within the Euler-safe regime regardless of the actual Δt chosen.

What carries the argument

The argument rests on rewriting the multistep multistage scheme as a nonstandard general linear method (replacing Δt with φ(Δt) in the standard GLM formulation), proving that consistency and stability are invariant under this substitution, and then using the Shu-Osher SSP decomposition to show that each internal stage is a convex combination of forward Euler steps whose effective step sizes are controlled by φ.

What would settle it

Find a convergent standard multistep multistage method of order p, choose φ satisfying φ(h) = h + O(h^{p+1}) and φ ≤ C·B_FE, and exhibit either (a) the nonstandard method failing to converge, (b) the nonstandard method dropping below order p, or (c) the nonstandard method violating the qualitative property for some positive Δt.

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Extended reading notes

Core claim

The substitution of Δt by a bounded function φ(Δt) in multistep multistage methods simultaneously preserves convergence, order of accuracy, and qualitative properties for all positive step sizes, provided φ matches the identity to sufficient Taylor order and is bounded by the SSP coefficient times the forward Euler stability limit.

Load-bearing premise

The preservation guarantee depends on the forward Euler method preserving the desired qualitative property for step sizes up to B_FE, and on φ being bounded by C times that limit. If the forward Euler bound is overly conservative or hard to compute for a given ODE, the practical bound on φ becomes correspondingly restrictive.

Editorial extensions

If this is right

  • Any convergent standard multistep multistage method can be converted into a nonstandard variant that preserves qualitative properties for all step sizes, without losing accuracy, by choosing φ appropriately.
  • The nonstandard Taylor theorem (Lemma 1) shows that the error analysis of nonstandard methods reduces to the standard Butcher tree / order condition theory with φ(Δt) replacing Δt, so existing order-condition tools transfer directly.
  • The sufficient bound C·B_FE on φ is empirically conservative; the paper's numerical experiments show the actual necessary bound B* can be significantly larger, suggesting tighter theoretical bounds are possible.
  • The framework extends naturally to implicit methods and to more general denominator functions φ(Δt, y_n), which could yield less restrictive bounds or higher accuracy.
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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

1 major / 7 minor

Summary. This paper introduces nonstandard versions of multistep multistage methods (in the Shu-Osher / SSP form) and, in the process, defines nonstandard general linear methods. The central results are threefold: (1) convergence of a nonstandard method is equivalent to convergence of its standard counterpart (Theorems 2-4), established via equivalence of consistency and stability; (2) the nonstandard method attains the same order p as the standard method if and only if the denominator function satisfies phi(h) = h + O(h^{p+1}) (Theorem 5, using a nonstandard Taylor theorem, Lemma 1); and (3) if phi is bounded by C * B_FE (where C is the SSP coefficient and B_FE is the forward Euler step-size bound), the method preserves qualitative properties such as non-negativity and boundedness for all positive step sizes (Theorem 6). Numerical experiments on a logistic equation and an SEIR system confirm the theoretical findings.

Significance. The paper provides a clean, self-contained extension of nonstandard finite difference theory to the general linear method framework, unifying prior work on nonstandard Runge-Kutta and nonstandard linear multistep methods. The theoretical results are correct and well-structured: the convergence equivalence follows from the observation that the coefficient matrices (V, U, B) are identical between standard and nonstandard forms; the order result is grounded in an independently proven nonstandard Taylor theorem (Lemma 1); and the preservation result correctly leverages the SSP convex-combination structure. The numerical experiments are thorough, testing both non-stiff and stiff regimes, multiple orders, and empirically comparing the sufficient bound C*B_FE against the necessary bound B*. The explicit acknowledgment that the sufficient bound can be conservative (Section 5.1.1) is a sign of intellectual honesty. The work is a natural and well-executed contribution to the nonstandard finite difference literature.

major comments (1)
  1. [Theorem 5, proof (Section 4)] The 'only if' direction of Theorem 5 (C3 implies C1) is argued in a single paragraph. The key claim is that the standard Taylor series of the error of the nonstandard method 'will contain at least one term in the form phi(Delta t) * d_0, where d_0 does not depend on Delta t,' and therefore phi(Delta t) = Delta t + O(Delta t^{p+1}) is necessary for order p. This argument is too compressed. For the standard method of order p, the leading error term is d_{p+1} * (Delta t)^{p+1}. When Delta t is replaced by phi(Delta t), the leading term becomes d_{p+1} * (phi(Delta t))^{p+1}, which is O(Delta t^{p+1}) whenever phi(Delta t) = O(Delta t) — a weaker condition than C1. The actual necessity of C1 must come from the first-order term: if phi(Delta t) = Delta t + c * Delta t^q + ... with q < p+1, then the first-order error term (phi(Delta t) - Delta t) * f(x(t)) = c * Delta t^q * f(x(t)) is generic
minor comments (7)
  1. [Section 2, Eq. (3)] The notation for the method coefficients alpha^{[n-l]}_{i,j} and beta^{[n-l]}_{i,j} uses a superscript [n-l] that could be confused with a power. A brief remark clarifying that [n-l] is an index (time-step label) would help readers unfamiliar with the Constantinescu-Sandu notation.
  2. [Remark 4 (Section 3.2)] The remark states that 'z can take any value on the complex plane' in the nonstandard case because phi is bounded but lambda is unrestricted. This is correct, but the phrasing 'all the classic results about the connection between the orders of general linear methods and the absolute stability also hold' could be misread as claiming that A-stability is achieved. A clarifying sentence noting that the stability region shape on the z-plane is identical (only the mapping z = lambda * phi(Delta t) differs) would prevent confusion.
  3. [Section 5.1.1, last paragraph] The bisection procedure for finding B* is described informally. A brief specification of the tolerance and the exact criterion for 'property is lost' (e.g., first timestep where boundedness is violated) would improve reproducibility.
  4. [Reference [42]] The title appears to have a typo: 'hallenges in Design Methods' should likely be 'Challenges in Design Methods'.
  5. [arXiv identifier] The arXiv identifier '2607.08694' corresponds to a July 2026 submission, which is consistent with the manuscript's references to 2025-2026 publications. This is fine but should be verified by the editorial office.
  6. [Section 4, proof of Lemma 1] The induction step uses R_{n-1} = (1/n!) f^{(n)}(a)(phi(delta),...,phi(delta)) + R_n, derived via partial integration. The argument is correct but the transition from the integral form to this recurrence could be spelled out in one additional line for clarity.
  7. [Appendix A] The SSP coefficients listed for the NGL methods (e.g., C=2.57 for NGLp2q2s3k3) are taken from [48]. It would be useful to note whether these are the optimal SSP coefficients or simply the values arising from the particular coefficient set chosen, so that readers interested in optimizing C can verify.

Circularity Check

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No significant circularity found

full rationale

The paper's three central results—convergence equivalence (Thm 3/4), order equivalence (Thm 5), and qualitative property preservation (Thm 6)—are derived from first principles within the SSP framework, not fitted to data or defined in terms of their own conclusions. The convergence proof follows the standard consistency+stability→convergence template for general linear methods (citing Butcher [50], an external source), with the key observation that the coefficient matrices V, U, B are identical between standard and nonstandard forms, making consistency and stability trivially equivalent (Props 1–2). The order result (Thm 5) relies on Lemma 1 (nonstandard Taylor theorem), which is proven independently by induction and partial integration, showing that replacing ∆t with φ(∆t) in the Taylor series preserves order conditions iff φ(h)=h+O(h^{p+1}). The preservation result (Thm 6) follows from the SSP structure: the method is a convex combination of forward Euler steps with effective step sizes (β/α)·φ(∆t), and the bound φ(∆t)≤C·B_FE ensures each forward Euler sub-step is within its stability bound. The consistency condition ∑α=1 makes the convex combination valid. The main self-citation is [44] (the author's prior work on nonstandard multistep methods), but it is used for context and comparison (e.g., noting similar behavior was observed for multistep methods), not as a load-bearing premise for the theorems. The standard multistep multistage method coefficients are cited from Constantinescu and Sandu [48], an independent source. No step in the derivation chain reduces to its inputs by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities or postulated objects. The function φ is a mathematical construct from prior nonstandard finite difference literature [17]. The SSP coefficient C is standard in SSP theory. The free parameters (φ choice, B_FE) are problem-dependent inputs, not fitted constants. The axioms are standard results from SSP theory and ODE analysis.

free parameters (2)
  • φ function choice = φ_p(x) = C·B_FE·x / ((C·B_FE)^p + x^p)^{1/p}
    The function φ is chosen by the practitioner to satisfy two conditions: φ(h)=h+O(h^{p+1}) for order p, and φ(x) ≤ C·B_FE for preservation. The specific form φ_p is recommended in Remark 7 but alternatives exist. C is the SSP coefficient of the underlying method (computed from the method coefficients), and B_FE is the forward Euler step-size bound for the specific ODE.
  • B_FE (forward Euler bound) = min(1/c, 1/x̃) for logistic; min(1/(γM), 1) for SEIR
    The maximum step size for which forward Euler preserves the desired property. This is problem-dependent and derived from the ODE structure, not fitted to data, but it is a free input parameter that determines the bound on φ.
assumptions (4)
  • standard math The standard multistep multistage method (2) in Shu-Osher form with nonnegative coefficients has SSP coefficient C = min{α/β}.
    This is the standard SSP theory from [45, 48], invoked in Section 3.3 to define the bound on φ.
  • domain assumption Conditions (P1)-(P3): existence of a closed invariant set S with bounded f and Lipschitz continuity.
    Stated in Section 2, used in the convergence proof (Theorem 2, Lemmas 4-6) to bound errors and ensure well-posedness.
  • domain assumption Covariance of the general linear method (shifting the input by η shifts the output by the same amount).
    Invoked before Theorem 1 in Section 3.1. Needed to prove that convergence implies consistency (Lemma 3).
  • domain assumption φ: R→R is continuously differentiable with φ(h) = h + O(h²) for small h.
    Stated in equation (4), Section 2. This is the minimal condition for the nonstandard method to be consistent; strengthened to φ(h)=h+O(h^{p+1}) for order p in Theorem 5.

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Pith. "Pith review of Some properties of high-order nonstandard multistep multistage methods." pith.science (2026). https://pith.science/paper/SOEXC25W

@misc{pith2026260708694,
  author       = {Pith},
  title        = {Pith review of: Some properties of high-order nonstandard multistep multistage methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SOEXC25W}},
  note         = {Machine review of arXiv:2607.08694}
}
read the original abstract

In this paper, we introduce nonstandard versions of multistep multistage methods. While proving the convergence of these schemes, we also define nonstandard general linear methods. We show that the nonstandard methods can attain the same order as their standard counterparts while preserving certain qualitative properties (e.g., boundedness) for all positive step sizes. These results are also demonstrated by some numerical experiments.

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Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [44]

    Anguelov, J

    R. Anguelov, J. M.-S. Lubuma, Contributions to the mathematics of the non- standard finite difference method and applications, Num. Meth. Part. Diff. Eq. Intern. J. 17 (2001) 518–543. https://doi.org/10.1002/num.1025

    work page doi:10.1002/num.1025 2001

  2. [1]

    Hadjimichael, D

    Y. Hadjimichael, D. Ketchenson, L. L´ oczi, A. N´ emeth, Strong stability preserving explicit linear multistep methods with variable step size, SIAM J. Num. Anal., 54(5) (2016) 2799–2832. https://doi.org/10.1137/15M101717X

    work page doi:10.1137/15m101717x 2016

  3. [2]

    N¨ ußlein, H

    S. N¨ ußlein, H. Ranocha, D. Ketcheson, Positivity-preserving adaptive Runge–Kutta methods, Com. Appl. Math. Comp. Sci., 16(2) (2021) 155–179. https://doi.org/10.2140/camcos.2021.16.155

    work page doi:10.2140/camcos.2021.16.155 2021

  4. [3]

    Ar´ evalo, G

    C. Ar´ evalo, G. S¨ oderlind, Y. Hadjimichael, I. Fekete, Local error estimation and step size control in adaptive linear multistep methods, Num. Alg. 86(2) (2021) 537–563. https://doi.org/10.1007/s11075-020-00900-1

    work page doi:10.1007/s11075-020-00900-1 2021

  5. [4]

    Charous, P

    A. Charous, P. Lermusiaux, Stable rank-adaptive dynamically orthogonal Runge–Kutta schemes, SIAM J. Sci. Comp. 46(1) (2025) A529–A560. https: //doi.org/10.1137/22M1534948 36 NSSPMS(4,2) NSSRK(2,2) NGLp2q2s3k3 ∆t errors orders errors orders errors orders 0.01 1.3039·10 −1 - 5.4342·10 −2 - 3.3848·10 −2 - 0.01·2 −1 4.0408·10 −2 1.6901 1.2925·10 −2 2.0719 1.0...

    work page doi:10.1137/22m1534948 2025

  6. [5]

    A. H. Morani, M. M. Saeed, M. Aslam, A. Mehmoud, A. Shokri, H. Mukalazi, Local and global stability analysis of HIV/AIDS by using a nonstandard finite difference scheme, Sci. Rep. 15(1) (2025) 4502. https://doi.org/10.1038/s41598-024-82872-z

    work page doi:10.1038/s41598-024-82872-z 2025

  7. [6]

    Zinihi, M

    A. Zinihi, M. Ehrhardt, M. R. S. Ammi, A nonstandard finite difference scheme for an SEIQR epidemiological PDE model, Appl. Math. Comp. 520 (2026) 129953. https://doi.org/10.1016/j.amc.2026.129953

    work page doi:10.1016/j.amc.2026.129953 2026

  8. [7]

    C. B. Marime, J. B. Munyakazi, A second-order nonstandard finite difference method for a malaria propagation model with control., AppliedMath 6(3) (2026)

    work page 2026

Show all 58 references
  1. [8]

    https://doi.org/10.3390/appliedmath6030036

    doi

  2. [9]

    A. J. O. Tass´ e, V. B. Kubalasa, B. Tsanou, Nonstandard finite difference schemes for some epidemic optimal control problems, Math. Comp. Sim. 228 (2025) 1–22. https://doi.org/10.1016/j.matcom.2024.08.028

    2025 doi

  3. [10]

    Wacker, Qualitative Study of a Dynamical System for Computer Virus Prop- agation—A Nonstandard Finite-Difference-Methodological View, Math

    B. Wacker, Qualitative Study of a Dynamical System for Computer Virus Prop- agation—A Nonstandard Finite-Difference-Methodological View, Math. Meth. Appl. Sci. 48(8) (2025) 9272–9291. https://doi.org/10.1002/mma.10798 37 NSSPMS(4,2) NSSRK(2,2) NGLp2q2s3k3 ∆t errors orders erro...

    2025 doi

  4. [11]

    M. T. Hoang, Mathematical analysis and numerical simulation of a generalized epidemiological model for malware propagation, Nonlin. Dyn. 114(1) (2026) 53. https://doi.org/10.1007/s11071-025-11912-8

    2026 doi

  5. [12]

    Faheem, B

    M. Faheem, B. Ghosh, Dynamics of a delayed discrete-time predator prey model proposed from a nonstandard finite difference scheme, J. Comp. Appl. Math. 458 (2025) 116346. https://doi.org/10.1016/j.cam.2024.116346

    2025 doi

  6. [13]

    Eskandari, Z

    Z. Eskandari, Z. Avazzadeh, R. Khoshsiar Ghaziani, B. Li, Dynamics and bifurcations of a discrete-time Lotka–Volterra model using nonstandard finite dif- ference discretization method, Math. Meth. Appl. Sci. 48(7) (2025) 7197–7212. https://doi.org/10.1002/mma.8859

    2025 doi

  7. [14]

    ¨Ozdo˘ gan, B

    N. ¨Ozdo˘ gan, B. Arslan, Nonstandard Finite Difference Theta Approaches to the Predator–Prey System, Math. Meth. Appl. Sci. 49(8) (2026) 8548–8561. https://doi.org/10.1002/mma.70486

    2026 doi

  8. [15]

    M. A. Mohye, J. B. Munyakazi, T. G. Dinka, Y. H. Haji, A. N. Ware, J. M. Ahmed, A new parameter-convergent nonstandard finite difference method for two-parameter singularly perturbed problems, Disc. Appl. Sci. 7(11) (2025) 1248. https://doi.org/10.1007/s42452-025-07721-8 38

    2025 doi

  9. [16]

    Fazayel, F

    M. Fazayel, F. Fakhar-Izadi, M. Dehghan, M. Abbaszadeh, Numerical simula- tion of Klein–Gordon–Zakharov equations using conservative nonstandard finite difference method combined with scalar auxiliary variable scheme, Comp. Appl. Math. 45(3) (2026) 110. https://doi.org/10.1007...

    2026 doi

  10. [17]

    Rehman, M

    B. Rehman, M. A. B. Iqbal, A. Khan, D. K. Almutairi, T. Abdeljawad, Non- standard Finite Difference Predictor Corrector Method for Quadratic Riccati Differential Equation, Eur. J. Pure Appl. Math. 18(2) (2025) 5703. https: //doi.org/10.29020/nybg.ejpam.v18i2.5703

    2025 doi

  11. [18]

    R. E. Mickens, Nonstandard finite difference models of differential equations, World Scientific, Singapore, 1993. https://doi.org/10.1142/2081

    1993 doi

  12. [19]

    D. T. Dimitrov, H. V. Kojouharov, Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems, Appl. Math. Lett. 18 (2005) 769–774. https://doi.org/10.1016/j.aml.2004.08.011

    2005 doi

  13. [20]

    D. T. Dimitrov, H. V. Kojouharov, Positive and elementary stable nonstandard numerical methods with applications to predator–prey models, J. Comp. Appl. Math. 189 (2006) 98–108. https://doi.org/10.1016/j.cam.2005.04.003

    2006 doi

  14. [21]

    Gupta, J

    M. Gupta, J. M. Slezak, F. K. Alalhareth, S. Roy, H. V. Kojouharov, Second- order modified nonstandard Runge-Kutta and theta methods for one-dimensional autonomous differential equations, Applic. Appl. Math. Intern. J. (AAM) 16 (2021) 1. https://digitalcommons.pvamu.edu/aam/vo...

    2021

  15. [22]

    H. V. Kojouharov, S. Roy, M. Gupta, F. Alalhareth, J. M. Slezak, A second-order modified nonstandard theta method for one-dimensional autonomous differential equations, Appl. Math. Lett. 112 (2021) 106775. https://doi.org/10.1016/j.aml. 2020.106775

    2021 doi

  16. [23]

    D. T. Dimitrov, H. V. Kojouharov, Stability-preserving finite-difference methods for general multi-dimensional autonomous dynamical systems, Int. J. Numer. Anal. Model 4 (2007) 282–292. https://www.math.ualberta.ca/ijnam/Volume-4- 2007/No-2-07/2007-02-06.pdf. (Accessed 8 July 2026)

    2007

  17. [24]

    Anguelov, P

    R. Anguelov, P. Kama, J. Lubuma, Nonstandard theta method and related discrete schemes for the reaction–diffusion equation, in: Proceedings of the Inter- national Conference of Computational Methods in Sciences and Engineering, World Scientific, Singapore, Vol. 1, 2003, pp. 24...

    2003

  18. [25]

    Anguelov, P

    R. Anguelov, P. Kama, J.-S. Lubuma, On non-standard finite difference models of reaction–diffusion equations, J. Comp. Appl. Math. 175 (2005) 11–29. https: //doi.org/10.1016/j.cam.2004.06.002 39

    2005 doi

  19. [26]

    J. M.-S. Lubuma, K. C. Patidar, Contributions to the theory of non-standard finite difference methods and applications to singular perturbation problems, in: Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2005, pp. 513–560. ...

    2005 doi

  20. [27]

    B. Wacker, Construction of High-Order Non-Standard Finite-Difference-Methods for Epidemiological Models, in: International Conference on Mathemat- ical Modeling in Physical Sciences, 693–706, 2024. https://doi.org/10.1007/ 978-3-032-00914-2 46

    2024

  21. [28]

    Bassenne, L

    M. Bassenne, L. Fu, A. Mani, Time-accurate and highly-stable explicit operators for stiff differential equations, J. Comp. Phys. 424 (2021) 109847. https://doi. org/10.1016/j.jcp.2020.109847

    2021 doi

  22. [29]

    Gonz´ alez-Parra, A

    G. Gonz´ alez-Parra, A. J. Arenas, B. M. Chen-Charpentier, Combination of nonstandard schemes and richardson’s extrapolation to improve the numeri- cal solution of population models, Math. Comp. Model. 52 (2010) 1030–1036. https://doi.org/10.1016/j.mcm.2010.03.015

    2010 doi

  23. [30]

    F. K. Alalhareth, M. Gupta, H. V. Kojouharov, S. Roy, Second-order modified nonstandard explicit euler and explicit runge–kutta methods for n-dimensional autonomous differential equations, Comp. 12 (2024) 183. https://doi.org/10. 3390/computation12090183

    2024

  24. [31]

    F. K. Alalhareth, M. Gupta, S. Roy, H. V. Kojouharov, Second-order modified positive and elementary stable nonstandard numerical methods for n-dimensional autonomous differential equations, Math. Meth. Appl. Sci. 48.7 (2025) 8037–8057. https://doi.org/10.1002/mma.9560

    2025 doi

  25. [32]

    Gupta, J

    M. Gupta, J. Slezak, F. Alalhareth, S. Roy, H. Kojouharov, Second-order non- standard explicit euler method, in: AIP Conference Proceedings, Vol. 2302, AIP Publishing, 2020. https://doi.org/10.1063/5.0033534

    2020 doi

  26. [33]

    M. T. Hoang, High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems, Num. Alg. (2024) 1–31. https://doi.org/10.1007/s11075-024-01792-1

    2024 doi

  27. [34]

    M. T. Hoang, M. Ehrhardt, A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model., J. Comp. Appl. Math. 444 (2024) 115752. https://doi.org/10.1016/j.cam.2024.115752

    2024 doi

  28. [35]

    Conte, G

    D. Conte, G. Pagano, T. Rold´ an, High order nonstandard finite-difference meth- ods, Appl. Math. Comp. 510 (2026) 129681. https://doi.org/10.1016/j.amc.2025. 129681 40

    2026 doi

  29. [36]

    Anguelov, J

    R. Anguelov, J. Lubuma, Second-order nonstandard finite difference schemes for a class of models in bioscience, arXiv preprint arXiv:2207.11618 (2022). https: //doi.org/10.48550/arXiv.2207.11618

    2022 doi arXiv

  30. [37]

    M. T. Hoang, A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems, Comm. Nonlin. Sci. Num. Sim. 114 (2022) 106654. https://doi.org/10.2139/ssrn.3958689

    2022 doi

  31. [38]

    B. M. Chen-Charpentier, D. T. Dimitrov, H. V. Kojouharov, Combined non- standard numerical methods for ODEs with polynomial right-hand sides, Math. Comp. Sim. 73 (2006) 105–113. https://doi.org/10.1016/j.matcom.2006.06.008

    2006 doi

  32. [39]

    H. V. Kojouharov, B. M. Chen-Charpentier, Nonstandard Eulerian–Lagrangian methods for multi-dimensional reactive transport problems, Appl. Num. Math. 49 (2004) 225–243. https://doi.org/10.1016/j.apnum.2002.04.001

    2004 doi

  33. [40]

    Mart´ ın-Vaquero, A

    J. Mart´ ın-Vaquero, A. Queiruga-Dios, A. M. del Rey, A. H. Encinas, J. H. Guillen, G. R. Sanchez, Variable step length algorithms with high-order extrapolated non- standard finite difference schemes for a SEIR model, J. Comp. Appl. Math. 330 (2018) 848–854. https://doi.org/10...

    2018 doi

  34. [41]

    A. H. Ejere, A nonstandard finite difference scheme for singularly perturbed reaction–diffusion differential equations with large negative and advance shifts, Math. Open 5 (2026) 2550021. https://doi.org/10.1142/s281100722550021x

    2026 doi

  35. [42]

    Q. A. Dang, M. T. Hoang, Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems, Intern. J. Comp. Math. 97 (2020) 2036–2054. https://doi.org/10.1080/00207160.2019. 1677895

    2020 doi

  36. [43]

    Jacques Periau, Springer Nature Switzerland, 2026, 179–194

    Farag´ o, I., Mosleh, R., Convergence Analysis of the Explicit Nonstandard Runge-Kutta Methods, in: hallenges in Design Methods, Numerical Tools and Technologies for Sustainable Aviation, Transport and Industry: Commemorative publication dedicated to the 80th Jubilee of Prof. ...

    2026 doi

  37. [45]

    Takacs An insight on some properties of high order nonstandard linear multi- step methods, Math

    B. Takacs An insight on some properties of high order nonstandard linear multi- step methods, Math. Comp. Sim. 245 (2026) 337–365. https://doi.org/10.1016/ j.matcom.2026.01.015

    2026

  38. [46]

    Gottlieb, D

    S. Gottlieb, D. Ketcheson, C. W. Shu, Strong stability preserving Runge-Kutta and multistep time discretizations, World Scientific, Singapore, 2011. https://doi. org/10.1142/7498 41

    2011 doi

  39. [47]

    Anguelov, J

    R. Anguelov, J. M.-S. Lubuma, F. Minani, Total variation diminishing nonstan- dard finite difference schemes for conservation laws, Math. Comp. Model. 51 (2010) 160–166. https://doi.org/10.1016/j.mcm.2009.08.038

    2010 doi

  40. [48]

    Mehdizadeh Khalsaraei, Nonstandard explicit third-order runge-kutta method with positivity property, Intern

    M. Mehdizadeh Khalsaraei, Nonstandard explicit third-order runge-kutta method with positivity property, Intern. J. Nonlin. Anal. Appl. 8 (2017) 37–46. https: //ijnaa.semnan.ac.ir/article 480.html (Accessed 8 July 2026)

    2017

  41. [49]

    E. M. Constantinescu, A. Sandu, Optimal explicit strong-stability-preserving general linear methods, SIAM J. Sci. Comp. 32.5 (2010) 3130–3150. https://doi. org/10.1137/090766206

    2010 doi

  42. [50]

    R. P. Agarwal, V. Lakshmikantham, Uniqueness and nonuniqueness criteria for ordinary differential equations, World Scientific, 1993. https://doi.org/10.1142/ 1988

    1993

  43. [51]

    J. C. Butcher, Numerical methods for ordinary differential equations, 2nd ed., John Wiley & Sons, 2016. https://doi.org/10.1002/9780470753767

    2016 doi

  44. [52]

    W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Royal Soc. London. Series A. 115 (1927) 700–721. https: //doi.org/10.1098/rspa.1927.0118

    1927 doi

  45. [53]

    Capasso, Mathematical structures of epidemic systems, Vol

    V. Capasso, Mathematical structures of epidemic systems, Vol. 97, Springer, Berlin, 1993. https://doi.org/10.1007/978-3-540-70514-7

    1993 doi

  46. [54]

    B. M. Takacs, G. S. Sebestyen, I. Farago, High-order reliable numerical methods for epidemic models with non-constant recruitment rate, Appl. Num. Math. 206 (2024) 75–93. https://doi.org/10.1016/j.apnum.2024.08.008

    2024 doi

  47. [55]

    Shu, Total-variation-diminishing time discretizations, SIAM J

    C.-W. Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comp. 9 (1988) 1073–1084. https://doi.org/10.1137/0909073

    1988 doi

  48. [56]

    Ketcheson, Computation of optimal monotonicity preserving general lin- ear methods, Math

    D. Ketcheson, Computation of optimal monotonicity preserving general lin- ear methods, Math. Comp. 78 (2009) 1497–1513. https://doi.org/10.1090/ s0025-5718-09-02209-1

    2009

  49. [57]

    Ketcheson, High order strong stability preserving time integrators and numer- ical wave propagation methods for hyperbolic PDEs, University of Washington, Seattle, 2009

    D. Ketcheson, High order strong stability preserving time integrators and numer- ical wave propagation methods for hyperbolic PDEs, University of Washington, Seattle, 2009. https://faculty.washington.edu/rjl/students/ketcheson/thesis.pdf (Accessed 8 July 2026)

    2009

  50. [58]

    Gottlieb, C.-W

    S. Gottlieb, C.-W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp. 67 (1998) 73–85. https://doi.org/10.1090/s0025-5718-98-00913-2 42

    1998 doi

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