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arxiv: 2604.17606 · v1 · submitted 2026-04-19 · 🧮 math.NA · cs.NA

Fully discrete scheme for the fifth-order KdV-Burgers-Fisher equation using Strang splitting and Fourier collocation methods

Nurcan G\"uc\"uyenen Kaymak , Fatma Z\"urnac{\i}-Yeti\c{s} , Muaz Seydao\u{g}lu This is my paper

Pith reviewed 2026-05-10 05:14 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords fifth-order KdV-Burgers-Fisher equationStrang splittingFourier collocationconvergence analysisnumerical PDEspectral methodsoperator splitting
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The pith

A fully discrete scheme using Strang splitting and Fourier collocation achieves second-order temporal and spectral spatial convergence for the fifth-order KdV-Burgers-Fisher equation.

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

The paper introduces a numerical method for the fifth-order Korteweg-de Vries-Burgers-Fisher equation by splitting the problem into linear and nonlinear subproblems solved via Strang splitting. Spatial discretization employs the Fourier collocation method, which provides spectral accuracy when the solution is sufficiently regular. Convergence analysis in Sobolev spaces establishes second-order accuracy in time using operator-theoretic arguments and Lie commutator estimates, with global errors derived from the Lady Windermere's fan argument. This combination allows efficient computation while maintaining high accuracy for modeling reaction, dissipative, and dispersive effects.

Core claim

The authors construct a fully discrete scheme for the fifth-order KdV-Burgers-Fisher equation by applying Strang splitting to separate linear and nonlinear operators and using Fourier collocation for space. They prove local error bounds via Banach space operator arguments and commutator estimates, then global second-order convergence in time with spectral accuracy in space in H^s using the Lady Windermere's fan argument. Numerical experiments confirm these rates.

What carries the argument

Strang splitting for temporal discretization of the split linear and nonlinear operators combined with Fourier collocation for spatial discretization.

If this is right

  • The scheme achieves second-order convergence in time.
  • The scheme achieves spectral convergence in space.
  • Numerical results confirm the theoretical error estimates and demonstrate the accuracy of the scheme.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could extend to other high-order nonlinear dispersive equations with similar operator structures.
  • The spectral accuracy in space suggests potential for long-time simulations with controlled dispersion errors.
  • Implementing the subproblems separately may reduce computational cost compared to unsplit methods.

Load-bearing premise

The solution possesses enough smoothness and regularity for the Fourier collocation method to attain spectral accuracy in space.

What would settle it

A counterexample or numerical experiment where the observed spatial convergence rate falls below spectral for a smooth solution would disprove the spectral accuracy claim.

Figures

Figures reproduced from arXiv: 2604.17606 by Fatma Z\"urnac{\i}-Yeti\c{s}, Muaz Seydao\u{g}lu, Nurcan G\"uc\"uyenen Kaymak.

Figure 1
Figure 1. Figure 1: Solution error versus the number of steps at t = 1 [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
read the original abstract

Operator splitting is an effective technique for the numerical solution of nonlinear partial differential equations by decomposing a complex problem into simpler subproblems. In this study, we present and analyze a fully discrete scheme for the fifth-order Korteweg-de Vries-Burgers-Fisher equation (KBF) by combining Strang splitting for time discretization with the Fourier collocation method for spatial discretization. In particular, the Fourier collocation method is an essential component of the proposed fully discrete scheme and yields spectral accuracy in space under suitable regularity assumptions. The KBF equation describes the interaction of reaction, dissipative, and dispersive mechanisms by incorporating the Fisher reaction term together with Burgers-type diffusion and higher-order KdV dispersion. The equation is split into a linear operator and a nonlinear operator, and the resulting subproblems are solved within the Strang splitting framework. Convergence is analyzed in the Sobolev space $H^s$. The local error is derived using operator-theoretic arguments in Banach spaces together with Lie commutator estimates, while the global error is obtained using the Lady Windermere's fan argument. The analysis yields second-order convergence in time and spectral convergence in space. Numerical results confirm the theoretical error estimates and demonstrate the accuracy of the proposed fully discrete scheme.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript develops a fully discrete scheme for the fifth-order KdV-Burgers-Fisher equation by combining Strang splitting for time discretization with Fourier collocation for spatial discretization. Convergence is analyzed in the Sobolev space H^s: local truncation errors are derived via operator-theoretic arguments in Banach spaces together with Lie commutator estimates, global errors follow from the Lady Windermere's fan argument, and the analysis concludes second-order accuracy in time together with spectral accuracy in space under suitable regularity assumptions on the solution. Numerical experiments are presented to confirm the predicted rates.

Significance. If the regularity assumptions are satisfied, the work supplies a practical high-order method for a nonlinear PDE that couples reaction, dissipation, and higher-order dispersion. The operator-theoretic treatment of the splitting error and the explicit numerical validation of both temporal and spatial orders are clear strengths; the approach is potentially reusable for related dispersive-reaction equations.

major comments (1)
  1. [Abstract and convergence theorem (presumably §4)] Abstract and convergence theorem (presumably §4): the spectral spatial convergence claim is stated to hold 'under suitable regularity assumptions,' yet the manuscript contains no regularity theorem or reference establishing that solutions of the fifth-order KBF equation remain in a class (e.g., Gevrey or analytic) that guarantees faster-than-polynomial decay of Fourier coefficients for all t>0. Fourier collocation yields only algebraic convergence in H^s for fixed s; this assumption is load-bearing for the spatial part of the main result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The single major comment raises an important point about the justification of regularity assumptions underlying the spectral spatial convergence claim. We address it directly below and will incorporate the necessary clarification in the revised version.

read point-by-point responses

  1. Referee: Abstract and convergence theorem (presumably §4): the spectral spatial convergence claim is stated to hold 'under suitable regularity assumptions,' yet the manuscript contains no regularity theorem or reference establishing that solutions of the fifth-order KBF equation remain in a class (e.g., Gevrey or analytic) that guarantees faster-than-polynomial decay of Fourier coefficients for all t>0. Fourier collocation yields only algebraic convergence in H^s for fixed s; this assumption is load-bearing for the spatial part of the main result.

    Authors: We agree that the spectral accuracy of Fourier collocation is conditional on the solution belonging to a regularity class (analytic or Gevrey) that produces sufficiently rapid decay of Fourier coefficients, whereas fixed-s Sobolev regularity would yield only algebraic rates. Our convergence theorem in Section 4 is explicitly stated under these 'suitable regularity assumptions,' and the local truncation error analysis via Lie commutators and the subsequent global error bound via Lady Windermere's fan argument both rely on them. The manuscript does not contain a dedicated regularity theorem or explicit reference establishing persistence of such regularity for the fifth-order KBF equation. In the revised manuscript we will add a short remark (near the statement of the main theorem and in the introduction) citing known well-posedness and Gevrey-regularity results for KdV-Burgers-type equations with lower-order nonlinear terms; we will also clarify that the spectral rate holds whenever the exact solution satisfies the requisite Fourier decay, which is consistent with the smooth initial data used in our numerical experiments. This addition will make the load-bearing assumption explicit without altering the core analysis. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent operator estimates and standard arguments

full rationale

The paper's convergence analysis relies on abstract operator-theoretic local error estimates via Lie commutator bounds in Banach spaces, followed by the Lady Windermere's fan argument to obtain global error bounds. These are general mathematical tools applied to the Strang splitting and Fourier collocation discretization; they do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. Spectral convergence in space is explicitly conditioned on 'suitable regularity assumptions' rather than derived tautologically. No uniqueness theorems, ansatzes, or renamings of known results are smuggled in via self-reference in the given text. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The scheme rests on standard functional-analytic properties of Fourier collocation and Strang splitting in Sobolev spaces; no free parameters, ad-hoc constants, or new postulated entities are introduced.

axioms (2)
  • standard math Fourier collocation yields spectral accuracy for sufficiently smooth periodic functions in H^s
    Invoked to obtain spatial error estimates under the regularity assumptions stated in the abstract.
  • standard math Strang splitting produces second-order local error via Lie commutator estimates in Banach spaces
    Used for the temporal error analysis.

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Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Awonusika, A.S

    R.O. Awonusika, A.S. Oke, Numerical solutions of nonline ar fifth-order KdV-type equations: Shifted Legendre collocation method, Int. J. Appl. Comput. Math. 12 (2026) Ar ticle 1. https://doi.org/10.1007/s40819-025-02074- 7. 20 N. Gücüyenen Kaymak et al

    work page doi:10.1007/s40819-025-02074- 2026

  2. [2]

    Khoshaim, M

    B. Khoshaim, M. Naeem, A. Akgul, N. Ghanmi, S. Zaland, Nove l analysis of fractional-order fifth-order Korteweg- de Vries equations, J. Math. 2022 (2022) 1883268. https://d oi.org/10.1155/2022/1883268

    work page doi:10.1155/2022/1883268 2022

  3. [3]

    Arora, R

    S. Arora, R. Jain, V.K. Kukreja, Solution of Benjamin-Bon a-Mahony-Burgers equation using collocation method with quintic Hermite splines, Appl. Numer. Math. 154 (2020) 1-16. https://doi.org/10.1016/j.apnum.2020.03.015

    work page doi:10.1016/j.apnum.2020.03.015 2020

  4. [4]

    Arora, R

    S. Arora, R. Mittal, B. Singh, Numerical solution of BBM-B urgers equation with quartic B-spline collocation method, J. Eng. Sci. Technol. 9 (2014) 104-116

    work page 2014

  5. [5]

    Hussain, S

    M. Hussain, S. Haq, Numerical solutions of strongly nonli near generalized Burgers-Fisher equation via meshfree spectral technique, Int. J. Comput. M ath. 98(9) (2021) 1727-1748. https://doi.org/10.1080/00207160.2020.1846729

    work page doi:10.1080/00207160.2020.1846729 2021

  6. [6]

    Karaagac, A

    B. Karaagac, A. Esen, K.M. Owolabi, E. Pindza, A trigonome tric quintic B-spline basis collocation method for the KdV-Kawahara equation, Numer. Anal. Appl. 16 (2023) 216-22 8. https://doi.org/10.1134/S1995423923030035

    work page doi:10.1134/s1995423923030035 2023

  7. [7]

    Karaagac, A

    B. Karaagac, A. Esen, K.M. Owolabi, E. Pindza, A collocati on method for solving time fractional nonlinear Korteweg-de Vries-Burgers equation arising in shallow wat er waves, Int. J. Mod. Phys. C 34 (2023) 2350096. https://doi.org/10.1142/S0129183123500961

    work page doi:10.1142/s0129183123500961 2023

  8. [8]

    Karaagac, A

    B. Karaagac, A. Esen, Y. Ucar, N.M. Yagmurlu, A trigonomet ric quintic B-spline collocation technique for the fifth-order KdV-Burgers-Fisher equation, Eng. Comput. 41 ( 2025) 4155-4171. https://doi.org/10.1007/s00366-024- 02049-7

    work page doi:10.1007/s00366-024- 2025

  9. [9]

    Q. Yang, H. Zhang, On the exact soliton solutions of fifth-o rder Korteweg-de Vries equation for surface gravity waves, Results Phys. 26 (2021) 104424. https://doi.org/10 .1016/j.rinp.2021.104424

    work page arXiv 2021

  10. [10]

    1968, SIAM Journal on Numerical Analysis, 5, 506, doi: 10.1137/0705041

    G. Strang, On the construction and comparison of differen ce schemes, SIAM J. Numer. Anal. 5 (3) (1968) 506-517. https://doi.org/10.1137/0705041

    work page doi:10.1137/0705041 1968

  11. [11]

    Marchuk, Methods of splitting, Nauka, Moscow, 1988

    G.I. Marchuk, Methods of splitting, Nauka, Moscow, 1988

    work page 1988

  12. [12]

    Jahnke, C

    T. Jahnke, C. Lubich, Error bounds for exponential opera tor splittings, BIT Numer. Math. 40 (2000) 735-744. https://doi.org/10.1023/a:1022396519656

    work page doi:10.1023/a:1022396519656 2000

  13. [13]

    Karta, Numerical solution for Benjamin-Bona-Mahony -Burgers equation with Strang time-splitting technique, Turkish J

    M. Karta, Numerical solution for Benjamin-Bona-Mahony -Burgers equation with Strang time-splitting technique, Turkish J. Math. 47 (2023) 537-553. https://doi.org/10.55 730/1300-0098.3377

    work page arXiv 2023

  14. [14]

    Gücüyenen, Strang splitting method to Benjamin-Bona -Mahony type equations: Analysis and application, J

    N. Gücüyenen, Strang splitting method to Benjamin-Bona -Mahony type equations: Analysis and application, J. Comput. Appl. Math. 318 (2017) 616-623. https://doi.org/1 0.1016/j.cam.2015.11.015

    work page 2017

  15. [15]

    Kuang, S

    B. Kuang, S. Xie, H. Fu, Strang splitting structure-pres erving high-order compact difference schemes for nonlinear convection diffusion equations, Commun. Nonl inear Sci. Numer. Simul. 146 (2025) 108749. https://doi.org/10.1016/j.cnsns.2025.108749

    work page doi:10.1016/j.cnsns.2025.108749 2025

  16. [16]

    Zürnacı, N.G

    F. Zürnacı, N.G. Kaymak, M. Seydaoğlu, G. Tanoğlu, Conve rgence analysis and numerical solution of the Benjamin-Bona-Mahony equation by Lie-Trotter split ting, Turkish J. Math. 42 (2018) 1471-1483. https://doi.org/10.3906/mat-1603-94

    work page doi:10.3906/mat-1603-94 2018

  17. [17]

    Zürnacı, M

    F. Zürnacı, M. Seydaoğlu, On the convergence of operator splitting for the Rosenau-Burgers equation, Numer. Methods Partial Differ. Equ. 35 (2019) 1363-1382. https://d oi.org/10.1002/num.22354

    work page doi:10.1002/num.22354 2019

  18. [18]

    Zürnacı-Yetiş, M

    F. Zürnacı-Yetiş, M. Seydaoğlu, Analysis of operator sp litting methods for the dispersive-Fisher equation, J. Math. Anal. Appl. 558 (2026) 130382. https://doi.org/10.1 016/j.jmaa.2025.130382

    work page arXiv 2026

  19. [19]

    Lubich, On splitting methods for Schrödinger-Poisso n and cubic nonlinear Schrödinger equations, Math

    C. Lubich, On splitting methods for Schrödinger-Poisso n and cubic nonlinear Schrödinger equations, Math. Comp. 77 (2008) 2141-2153. https://doi.org/10.1090/s0025-571 8-08-02101-7

    work page doi:10.1090/s0025-571 2008

  20. [20]

    Holden, C

    H. Holden, C. Lubich, H. Risebro, Operator splitting for partial differential equations with Burgers nonlinearity, Math. Comp. 82 (2013) 173-185.https://doi.org/10.1090/s 0025-5718-2012-02624-x

    work page doi:10.1090/s 2013

  21. [21]

    Bulut, O

    F. Bulut, O. Oruc, A. Esen, Higher order Haar wavelet meth od integrated with Strang split- ting for solving regularized long wave equation, Math. Comp ut. Simul. 197 (2022) 277-290. https://doi.org/10.1016/j.matcom.2022.02.006

    work page doi:10.1016/j.matcom.2022.02.006 2022

  22. [22]

    Gauckler, Convergence of a split-step Hermite method for the Gross-Pitaevskii equation, IMA J

    L. Gauckler, Convergence of a split-step Hermite method for the Gross-Pitaevskii equation, IMA J. Numer. Anal. 31 (2011) 396-415. https://doi.org/10.1093/imanum/drp0 41

    work page doi:10.1093/imanum/drp0 2011

  23. [23]

    Lubich, From quantum to classical molecular dynamics : Reduced models and numerical analysis, EMS Press, Zurich, 2008

    C. Lubich, From quantum to classical molecular dynamics : Reduced models and numerical analysis, EMS Press, Zurich, 2008. https://doi.org/10.4171/067

    work page doi:10.4171/067 2008

  24. [24]

    Gücüyenen Kaymak, Error estimates of the Fourier coll ocation combined Strang splitting method for Benjamin-Bona-Mahony type equations, Turkish J

    N. Gücüyenen Kaymak, Error estimates of the Fourier coll ocation combined Strang splitting method for Benjamin-Bona-Mahony type equations, Turkish J . Math. Comput. Sci. 18 (2026) 78-94. https://doi.org/10.47000/tjmcs.1674288

    work page doi:10.47000/tjmcs.1674288 2026

  25. [25]

    O. Oruc, A. Esen, F. Bulut, A Strang splitting approach co mbined with Chebyshev wavelets to solve the regular- ized long-wave equation numerically, Mediterr. J. Math. 17 (2020) 40. https://doi.org/10.1007/s00009-020-01572- w

    work page doi:10.1007/s00009-020-01572- 2020

  26. [26]

    Xavier, M.A

    J.C. Xavier, M.A. Rincon, D.G. Alfaro Vigo, Error estima tes for a fully discrete spectral scheme for nonlinear Boussinesq systems, Numer. Math. 149 (2021) 679-716. https ://doi.org/10.1007/s00211-021-01239-y

    work page doi:10.1007/s00211-021-01239-y 2021

  27. [27]

    C. Liu, C. Wang, Y. Wang, A structure-preserving operato r splitting scheme for reaction-diffusion equations with detailed balance, J. Comput. Phys. 436 (2021) 110253. https ://doi.org/10.1016/j.jcp.2021.110253. Fully discrete scheme for the fifth-order KdV-Burgers-Fish er equation 21

    work page doi:10.1016/j.jcp.2021.110253 2021

  28. [28]

    C. Liu, C. Wang, Y. Wang, A second-order accurate operato r splitting scheme for reaction-diffusion systems in an energetic variational formulation, SIAM J. Sc i. Comput. 44 (2022) A2276-A2301. https://doi.org/10.1137/21M1444825

    work page doi:10.1137/21m1444825 2022

  29. [29]

    C. Liu, C. Wang, Y. Wang, S.M. Wise, Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balanc e, SIAM J. Numer. Anal. 60 (2022) 781-803. https://doi.org/10.1137/21M1421283

    work page doi:10.1137/21m1421283 2022

  30. [30]

    Zhang, H

    C. Zhang, H. Wang, J. Huang, C. Wang, X. Yue, A second-orde r operator splitting nu- merical scheme for the "good" Boussinesq equation, Appl. Nu mer. Math. 119 (2017) 179-193. https://doi.org/10.1016/j.apnum.2017.05.006

    work page doi:10.1016/j.apnum.2017.05.006 2017

  31. [31]

    2000 , publisher =

    L.N. Trefethen, Spectral methods in MATLAB, SIAM, Phila delphia, 2000. https://doi.org/10.1137/1.9780898719598

    work page doi:10.1137/1.9780898719598 2000