Fourth-order accurate difference schemes for solving Benjamin–Bona–Mahony–Burgers (BBMB) equation

Abstract

In this paper, two high-order difference schemes for the Benjamin–Bona–Mahony–Burgers (BBMB) equation are proposed. The first scheme is two level and nonlinear implicit, the second scheme is three level and linear implicit. A priori estimates for the numerical solution are derived. It is proved that the difference schemes are uniquely solvable and unconditionally convergent, in discrete maximum norm, with the convergence order of two in time and four in space. Numerical experiments are given to show the efficiency and accuracy of our methods.

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References

  1. 1.

    Benjamin RT, Bona JL, Mahony JJ (1972) Model equations for long waves in nonlinear dispersive systems. Philos Trans R Soc Lond 272:47–78

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bona JL (1978) Model equations for waves in nonlinear dispersive systems. In: Proceedings of the international congress of mathematicians, Helsinki

  3. 3.

    Peregrine DH (1996) Calculations of the development of an undular bore. J Fluid Mech 25:321–326

    Article  Google Scholar 

  4. 4.

    Omrani K, Ayadi M (2008) Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation. Numer Methods Partial Differ Equ 24(1):239–248

    MathSciNet  Article  Google Scholar 

  5. 5.

    Achouri T, Khiari N, Omrani K (2006) On the convergence of difference schemes for the Benjamin-Bona-Mahony (BBM) equation. Appl Math Comput 182(2):999–1005

    MathSciNet  MATH  Google Scholar 

  6. 6.

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

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 8.

    Berikelashvili G, Mirianashvili M (2011) A one-parameter family of difference schemes for the regularized long-wave equation. Georgian Math J 18:639–667

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kutluay S, Esen A (2006) A finite difference solution of the regularized long-wave equation. Math Probl Eng 2006:1–14

    MathSciNet  Article  Google Scholar 

  10. 10.

    Rashid A (2005) A three levels finite difference method for the nonlinear regularized long wave equation. Mem Differ Equ Math Phys 34:135–146

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Kadri T, Khiari N, Abidi F, Omrani K (2008) Methods for the Numerical Solution of the Benjamin-Bona-Mahony-Burgers Equation. Numer Methods Partial Differ Equ 24(6):1501–1516

    MathSciNet  Article  Google Scholar 

  12. 12.

    Omrani K (2006) The convergence of fully discrete Galerkin approximations for the Benjamin-Bona-Mahony (BBM) equation. Appl Math Comput 180(2):614–621

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Achouri T, Ayadi M, Omrani K (2009) A fully Galerkin method for the damped generalized regularized long-wave (DGRLW) equation. Numer Methods Partial Differential Equ 25:668–684

    MathSciNet  Article  Google Scholar 

  14. 14.

    Dogan A (1997) Petrov-Galerkin finite element methods. Thesis Phil, Doct

  15. 15.

    Raslan KR (2005) A computational method for the regularized long wave (RLW) equation. Appl Math Comput 167(2):1101–1118

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Dag I, Özer MN (2001) Approximation of the RLW equation by the least square cubic B-spline finite element method. Appl Math Model 3:221–231

    Article  Google Scholar 

  17. 17.

    Achouri T, Omrani K (2009) Numerical solutions for the damped generalized regularized long-wave equation with a variable coefficient by Adomian decomposition method. Commun Nonlinear Sci Numer Simulat 14:2025–2033

    MathSciNet  Article  Google Scholar 

  18. 18.

    Labidi M, Omrani K (2011) Numerical simulation of the modified regularized long wave equation by He’s variational iteration method. Numer Methods Partial Differ Equ 27:478–489

    Article  Google Scholar 

  19. 19.

    Achouri T, Omrani K (2010) Application of the homotopy perturbation method to the modified regularized long-wave equation. Numer Methods Partial Differ Equ 26(2):399–411

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

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

    Article  MATH  Google Scholar 

  22. 22.

    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 

  23. 23.

    Noureddine Atouani, Omrani Khaled (2014) A new conservative high-order accurate difference scheme for the Rosenau equation. Appl Anal 94(12):1–21

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Ghiloufi A, Rahmeni M, Omrani K (2019) Convergence of two conservative high-order accurate difference schemes for the generalized Rosenau-Kawahara-RLW equation. Eng Comput. https://doi.org/10.1007/s00366-019-00719-y

    Article  Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

    Browder FE (1965) Existence and uniqueness theorems for solutions of nonlinear boundary value problems, In: Finn R, (eds) Applications of nonlinear P.D.Es. proceedings of symposium of applied mathematics, vol. 17, A.M.S, Providence, pp 24–49

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Acknowledgements

I would like to thank the reviewers that their comments and suggestions have really improved the quality of the paper.

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Correspondence to Khedidja Bayarassou.

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Bayarassou, K. Fourth-order accurate difference schemes for solving Benjamin–Bona–Mahony–Burgers (BBMB) equation. Engineering with Computers 37, 123–138 (2021). https://doi.org/10.1007/s00366-019-00812-2

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Keywords

  • BBMB equation
  • Nonlinear difference scheme
  • Linearized difference scheme
  • Fourth-order accuracy
  • Unique solvability
  • Convergence