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A three-level linearized high-order accuracy difference scheme for the extended Fisher–Kolmogorov equation

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Abstract

A three-level linearized difference scheme for the extended Fisher–Kolmogorov equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergence order in maximum norm is \(O(h^4+k^2)\), where k and h are the temporal and spatial grid sizes, respectively. The accurateness and effectiveness of the method are tested by taking various examples. The numerical results of the method are compared with the exact solutions and also compared with earlier published results. It is found that the proposed method produces more accurate results than the others available in the literature.

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References

  1. Coullet P, Elphick C, Repaux D (1987) Nature of spatial chaos. Phys Rev Lett 58:431–434

    MathSciNet  Google Scholar 

  2. Dee GT, van Saarloos W (1988) Bistable systems with propagating fronts leading to pattern formation. Phys Rev Lett 60:2641–2644

    Google Scholar 

  3. Zhu G (1982) Experiments on director waves in nematic liquid crystals. Phys Rev Lett 49:1332–1335

    Google Scholar 

  4. Ahlers G, Cannell DS (1983) Vortexfront propagation in rotating Couette Taylor Tow. Phys Rev Lett 50:1583–1586

    Google Scholar 

  5. Aronson DG, Weinberger HF (1978) Multidimensional nonlinear diJusion arising in population genetics. Adv Math 30:33–67

    MATH  Google Scholar 

  6. Hornreich RM, Luban M, Shtrikman S (1975) Critical behaviour at the onset of \(k\)-space instability at the \(\lambda\) line. Phys Rev Lett 35:1678–1681

    Google Scholar 

  7. Danumjaya P, Pani AK (2006) Numerical methods for the extended Fisher-Kolmogorov (EFK) equation. Int J Numer Anal Model 3(2):186–210

    MathSciNet  MATH  Google Scholar 

  8. Doss LJT, Nandini AP (2012) An \(H^1\)-Galerkin mixed finite element method for the extended Fisher-Kolmogorov equation. Int J Numer Anal Model 3(4):460–485

    MathSciNet  MATH  Google Scholar 

  9. Doss, L Jones Tarcius, Kousalya N (2020) A finite pointset method for Extended Fisher-Kolmogorov equation based on mixed formulation. Int J Comput Methods. https://doi.org/10.1142/S021987622050019X

  10. Liu F, Zhao X, Liu B (2017) Fourier pseudo-spectral method for the extended Fisher-Kolmogorov equation in two dimensions. Adv Diff Equ 2017(1):94

    MathSciNet  MATH  Google Scholar 

  11. Mittal R, Arora G (2010) Quintic B-spline collocation method for numerical solution of the extended Fisher-Kolmogorov equation. Int J Appl Math Mech 6(1):74–85

    Google Scholar 

  12. Ibrahim C (2020) Gegenbauer wavelet collocation method for the extended Fisher-Kolmogorov equation in two dimensions. Math Methods Appl Sci. https://doi.org/10.1002/mma.6300

  13. Oruc \(\ddot{\rm O}\) (2019) An efficient wavelet collocation method for nonlinear two-space dimensional Fisher-Kolmogorov-Petrovsky-Piscounov equation and two-space dimensional extended Fisher-Kolmogorov equation. Eng Comput 1–18. https://doi.org/10.1007/s00366-019-00734-z

  14. Abbaszadeh M, Dehghan M, Khodadadian A, Heitzinger C (2020) Error analysis of interpolating element free Galerkin method to solve non-linear extended Fisher-Kolmogorov equation. Comput Math Appl. https://doi.org/10.1016/j.camwa.2020.03.014

  15. Ilati M, Dehghan M (2017) Direct local boundary integral equation method for numerical solution of extended Fisher Kolmogorov equation. Eng Comput 34:203–213

    Google Scholar 

  16. Dehghan M, Shafieeabyaneh N (2019) Local radial basis function-finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher-Kolmogorov equations. Engineering with Computers. https://doi.org/10.1007/s00366-019-00877-z

  17. Kadri Tlili, Omrani Khaled (2011) A second-order accurate difference scheme for an extended Fisher-Kolmogorov equation. Comput Math Appl 61:451–459

    MathSciNet  MATH  Google Scholar 

  18. Khiari Noomen, Omrani Khaled (2011) Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions. Comput Math Appl 62(11):4151–4160

    MathSciNet  MATH  Google Scholar 

  19. Kadri Tlili, Omrani Khaled (2018) A fourth-order accurate finite difference scheme for the extended-Fisher-Kolmogorov equation. Bull Korean Math Soc 55(1):297–310

    MathSciNet  MATH  Google Scholar 

  20. He Dongdong (2016) On the \(L^{\infty }\)-norm convergence of a three-level linearly implicit finite difference method for the extended Fisher-Kolmogorov equation in both 1D and 2D. Comput Math Appl 71(12):2594–2607

    MathSciNet  MATH  Google Scholar 

  21. Karakoc SBG, Gao F, Bhowmik SK (2018) Solitons and shock waves solutions for the Rosenau-KdV-RLW equation. J Sci Arts 4(45):1073–1088

    Google Scholar 

  22. Ak T, Gazi Karakoç SB, Triki H (2016) Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation. Eur Phys J Plus Sayi 131:1–15

  23. Karakoc SBG, Ak T (2016) Numerical simulation of dispersive shallow water waves with Rosenau-KdV equation. Int J Adv Appl Math Mech 3:32–40

    MathSciNet  MATH  Google Scholar 

  24. Karakoc SBG (2018) A detailed numerical study on generalized Rosenau-KdV equation with finite element method. J Sci Arts 4(45):837–852

    Google Scholar 

  25. Omrani Khaled, Ayadi Mekki (2008) Finite difference discretization of the Benjamin-Bona-Mahony-Burgers (BBMB) equation. Numer Methods Partial Diff Equ 24(1):239–248

    MATH  Google Scholar 

  26. Rouatbi Asma, Omrani Khaled (2017) Two conservative difference schemes for a model of nonlinear dispersive equations. Chaos Solitons Fractals 104:516–530

    MathSciNet  MATH  Google Scholar 

  27. Rouatbi A, Achouri T, Omrani K (2018) High-order conservative difference scheme for a model of nonlinear dispersive equations. Comput Appl Math 37:4169–4195. https://doi.org/10.1007/s40314-017-0567-1

    Article  MathSciNet  MATH  Google Scholar 

  28. Ghiloufi A, Rouatbi A, Omrani K (2018) A new conservative fourth-order accurate difference scheme for solving a model of nonlinear dispersive equations. Math Methods Appl Sci 41:5230–5253. https://doi.org/10.1002/mma.5073

    Article  MathSciNet  MATH  Google Scholar 

  29. Ghiloufi A, Omrani K (2017) New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves. Numer Methods Partial Diff Equ 34:451–500. https://doi.org/10.1002/num.22208

    Article  MathSciNet  MATH  Google Scholar 

  30. Omrani K, Ghiloufi A (2020) An efficient computational approach for two-dimensional variant of nonlinear-dispersive model of shallow water wave, 2020. Eng Comput. https://doi.org/10.1007/s00366-020-00967-3

  31. He Dongdong (2016) Exact solitary solution and a three-level linearly implicit conservative finite difference method for the generalized Rosenau-Kawahara-RLW equation with generalized Novikov type perturbation. Nonlinear Dyn 85(1):479–498

    MathSciNet  MATH  Google Scholar 

  32. He Dongdong (2015) New solitary solutions and a conservative numerical method for the Rosenau-Kawahara equation with power law nonlinearity. Nonlinear Dyn 82:1177–1190

    MathSciNet  MATH  Google Scholar 

  33. Dehghan M, Taleei A (2010) A compact split-step finite difference method for solving the nonlinear Schrodinger equations with constant and variable coefficients. Comput Phys Commun 181:43–51

    MathSciNet  MATH  Google Scholar 

  34. Dehghan M, Mohebbi A, Asgari Z (2009) Fourth-order compact solution of the nonlinear Klein-Gordon equation. Numer Algorithms 52:523–540

    MathSciNet  MATH  Google Scholar 

  35. Mohebbi A, Dehghan M (2010) High-order solution of one-dimensional Sine-Gordon equation using compact finite difference and DIRKN methods. Math Comput Model 51:537–549

    MathSciNet  MATH  Google Scholar 

  36. Mohebbi A, Dehghan M (2010) High-order compact solution of the one-dimensional heat and advection-diffusion equations. Appl Math Model 34:3071–3084

    MathSciNet  MATH  Google Scholar 

  37. Karaa S, Zhang J (2004) High-order ADI method for solving unsteady convection-diffusion problems. J Comput Phys 198:1–9

    MathSciNet  MATH  Google Scholar 

  38. Karaa S (2006) High-order compact ADI method for solving three-dimentional unsteady convection diffusion problems. Numer Methods Partial Differ Equ 22:983–993

    MATH  Google Scholar 

  39. Karaa S (2007) High-order difference schemes for 2D elliptic and parabolic problems with mixed derivatives. Numer Methods Partial Differ Equ 23:366–378

    MathSciNet  MATH  Google Scholar 

  40. Karaa S (2007) High-order ADI method for stream-function vorticity equations. Proc Appl Math Mech 7:1025601–1025602

    Google Scholar 

  41. Dehghan M, Abbaszadeh M (2019) The simulation of some chemotactic bacteria patterns in liquid medium which arises in tumor growth with blow-up phenomena via a generalized smoothed particle hydrodynamics (GSPH) method. Eng Comput 35:875–892

    Google Scholar 

  42. Abbaszadeh M, Dehghan M (2019) Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed-order advection diffusion equation. Eng Comput (in press)

  43. Abbaszadeh M, Dehghan M (2020) CrankNicolson Galerkin spectral method for solving two-dimensional time-space distributed-order weakly singular integro-partial differential equation. J Comput Appl Math 374:112739

    MathSciNet  MATH  Google Scholar 

  44. Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71:16–30

    MathSciNet  MATH  Google Scholar 

  45. Abbaszadeh M, Dehghan M (2019) The solution of nonlinear Green-Naghdi equation arising in water sciences via a meshless method which combines moving kriging interpolation shape functions with the weighted essentially non-oscillatory method. Commun Nonlinear Sci Numer Simul 68:220–239

    MathSciNet  MATH  Google Scholar 

  46. Dehghan M, Abbaszadeh M, Mohebbi A (2015) The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J Comput Appl Math 286:211–231

    MathSciNet  MATH  Google Scholar 

  47. Zhou Y (1990) Application of discrete functional analysis to the finite difference methods. International Academic Publishers, Bijing

    Google Scholar 

  48. Omrani K (2004) On fully discrete Galerkin approximations for the Cahn-Hilliard equation. Math Model Anal 9(4):313–326

    MathSciNet  MATH  Google Scholar 

  49. Khiari N, Achouri T, Ben Mohamed ML, Omrani K (2007) Finite difference approximate solutions for the Cahn-Hilliard equation. Numer Methods Partial Differ Equ 23:437–455

    MathSciNet  MATH  Google Scholar 

  50. Omrani K (2003) A second-order splitting method for a finite difference scheme for the Sivashinsky equation. Appl Math Lett 16:441–445

    MathSciNet  MATH  Google Scholar 

  51. Omrani K, Ben Mohamed M (2005) A linearized difference scheme for the Sivashinsky equation. Far East J Appl Math 20:179–188

    MathSciNet  MATH  Google Scholar 

  52. Omrani K (2007) Numerical methods and error analysis for the nonlinear Sivashinsky equation. Appl Math Comput 189:949–962

    MathSciNet  MATH  Google Scholar 

  53. Atouani N, Omrani K (2015) On the convergence of conservative difference schemes for the 2D generalized Rosenau-Korteweg de Vries equation. Appl Math Comput 250:832–847

    MathSciNet  MATH  Google Scholar 

  54. Atouani N, Omrani K (2015) A new conservative high-order accurate difference scheme for the Rosenau equation. Appl Anal 94:2435–2455

    MathSciNet  MATH  Google Scholar 

  55. Omrani K, Abidi F, Achouri T, Khiari N (2008) A new conservative finite difference scheme for the Rosenau equation. Appl Math Comput 201:35–43

    MathSciNet  MATH  Google Scholar 

  56. Ghiloufi Ahlem, Rahmeni Mohamed, Omrani Khaled (2020) Convergence of two conservative high-order accurate difference schemes for the generalized Rosenau-Kawahara-RLW equation. Eng Comput 36:617–632

    Google Scholar 

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Authors and Affiliations

  1. Ecole Polytechnique de Tunis, Rue El Khawarezmi, La Marsa, 2078, Tunisia

    Kaouther Ismail

  2. Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, Université de Sousse, Sousse Ibn Khaldoun, Sousse, 4003, Tunisia

    Noureddine Atouani

  3. Institut Préparatoire aux Etudes d’Ingénieurs d’El Manar, Université de Tunis El Manar, Tunis, 2092, Tunisia

    Khaled Omrani

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  1. Kaouther Ismail
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  2. Noureddine Atouani
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Correspondence to Khaled Omrani.

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Ismail, K., Atouani, N. & Omrani, K. A three-level linearized high-order accuracy difference scheme for the extended Fisher–Kolmogorov equation. Engineering with Computers 38 (Suppl 2), 1215–1225 (2022). https://doi.org/10.1007/s00366-020-01269-4

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