Skip to main content
SpringerLink
Log in

A new energy conservative scheme for regularized long wave equation

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

An energy conservative scheme is proposed for the regularized long wave (RLW) equation. The integral method with variational limit is used to discretize the spatial derivative and the finite difference method is used to discretize the time derivative. The energy conservation of the scheme and existence of the numerical solution are proved. The convergence of the order O(h2 + τ2) and unconditional stability are also derived. Numerical examples are carried out to verify the correctness of the theoretical analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Banquet Brango: The symmetric regularized-long-wave equation: Well-posedness and nonlinear stability. Physica D 241 (2012), 125–133.

    Article  Google Scholar 

  2. D. Bhardwaj, R. Shankar: A computational method for regularized long wave equation. Comput. Math. Appl. 40 (2000), 1397–1404.

    Article  MathSciNet  Google Scholar 

  3. S. K. Bhowmik, S. B. G. Karakoc: Numerical approximation of the generalized regularized long wave equation using Petrov-Galerkin finite element method. Numer. Methods Partial Differ. Equations 35 (2019), 2236–2257.

    Article  MathSciNet  Google Scholar 

  4. J. Cai: Multisymplectic numerical method for the regularized long-wave equation. Comput. Phys. Commun. 180 (2009), 1821–1831.

    Article  MathSciNet  Google Scholar 

  5. N. G. Chegini, A. Salaripanah, R. Mokhtari, D. Isvand: Numerical solution of the regularized long wave equation using nonpolynomial splines. Nonlinear Dyn. 69 (2012), 459–471.

    Article  MathSciNet  Google Scholar 

  6. R. Chertovskih, A. C.-L. Chian, O. Podvigina, E. L. Rempel, V. Zheligovsky: Existence, uniqueness, and analyticity of space-periodic solutions to the regularized long-wave equation. Adv. Differ. Equ. 19 (2014), 725–754.

    MathSciNet  MATH  Google Scholar 

  7. A. Dogan: Numerical solution of RLW equation using linear finite elements within Galerkin’s method. Appl. Math. Modelling 26 (2002), 771–783.

    Article  Google Scholar 

  8. J. C. Eilbeck, G. R. McGuire: Numerical study of the regularized long-wave equation. I: Numerical methods. J. Comput. Phys. 19 (1975), 43–57.

    Article  MathSciNet  Google Scholar 

  9. S. Fang, B. Guo, H. Qiu: The existence of global attractors for a system of multi-dimensional symmetric regularized wave equations. Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 61–68.

    Article  MathSciNet  Google Scholar 

  10. L. R. T. Gardner, G. A. Gardner, I. Dag: A B-spline finite element method for the regularized long wave equation. Commun. Numer. Methods Eng. 11 (1995), 59–68.

    Article  MathSciNet  Google Scholar 

  11. L. Guo, H. Chen: H1-Galerkin mixed finite element method for the regularized long wave equation. Computing 77 (2006), 205–221.

    Article  MathSciNet  Google Scholar 

  12. B. Guo, Y. Shang: Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term. Acta Math. Appl. Sin., Engl. Ser. 19 (2003), 191–204.

    Article  MathSciNet  Google Scholar 

  13. D. A. Hammad, M. S. El-Azab: Chebyshev-Chebyshev spectral collocation method for solving the generalized regularized long wave (GRLW) equation. Appl. Math. Comput. 285 (2016), 228–240.

    MathSciNet  MATH  Google Scholar 

  14. J. Hu, K. Zheng: Two conservative difference schemes for the generalized Rosenau equation. Bound. Value Probl. 2010 (2010), Article ID 543503, 18 pages.

  15. D. Irk, P. Keskin: Quadratic trigonometric B-spline Galerkin methods for the regularized long wave equation. J. Appl. Anal. Comput. 7 (2017), 617–631.

    MathSciNet  MATH  Google Scholar 

  16. D. Irk, P. Keskin Yildiz, M. Zorşahin Görgülü: Quartic trigonometric B-spline algorithm for numerical solution of the regularized long wave equation. Turk. J. Math. 43 (2019), 112–125.

    Article  MathSciNet  Google Scholar 

  17. S. B. G. Karakoc, N. M. Yagmurlu, Y. Ucar: Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines. Bound. Value Probl. 2013 (2013), Article ID 27, 17 pages.

  18. R. Kumar, S. Baskar: B-spline quasi-interpolation based numerical methods for some Sobolev type equations. J. Comput. Appl. Math. 292 (2016), 41–66.

    Article  MathSciNet  Google Scholar 

  19. B. Lin: A nonpolynomial spline scheme for the generalized regularized long wave equation. Stud. Appl. Math. 132 (2014), 160–182.

    Article  MathSciNet  Google Scholar 

  20. B. Lin: Parametric spline solution of the regularized long wave equation. Appl. Math. Comput. 243 (2014), 358–367.

    MathSciNet  MATH  Google Scholar 

  21. B. Lin: Non-polynomial splines method for numerical solutions of the regularized long wave equation. Int. J. Comput. Math. 92 (2015), 1591–1607.

    Article  MathSciNet  Google Scholar 

  22. Y. Luo, X. Li, C. Guo: Fourth-order compact and energy conservative scheme for solving nonlinear Klein-Gordon equation. Numer. Methods Partial Differ. Equations 33 (2017), 1283–1304.

    Article  MathSciNet  Google Scholar 

  23. Ö. Oruç, F. Bulut, A. Esen: Numerical solutions of regularized long wave equation by Haar wavelet method. Mediterr. J. Math. 13 (2016), 3235–3253.

    Article  MathSciNet  Google Scholar 

  24. D. H. Peregrine: Calculations of the development of an undular bore. J. Fluid Mech. 25 (1966), 321–330.

    Article  Google Scholar 

  25. D. H. Peregrine: Long waves on a beach. J. Fluid Mech. 27 (1967), 815–827.

    Article  Google Scholar 

  26. E. Pindza, E. Maré: Solving the generalized regularized long wave equation using a distributed approximating functional method. Int. J. Comput. Math. 2014 (2014), Article ID 178024, 12 pages.

  27. K. R. Raslan: A computational method for the regularized long wave (RLW) equation. Appl. Math. Comput. 167 (2005), 1101–1118.

    MathSciNet  MATH  Google Scholar 

  28. A. Rouatbi, T. Achouri, K. Omrani: High-order conservative difference scheme for a model of nonlinear dispersive equations. Comput. Appl. Math. 37 (2018), 4169–4195.

    Article  MathSciNet  Google Scholar 

  29. H. Salih, L. N. M. Tawfiq, Z. R. Yahya, S. Mat Zin: Solving modified regularized long wave equation using collocation method. J. Phys., Conf. Ser. 1003 (2018), Article ID 012062, 9 pages.

  30. Y. Shang, B. Guo: Exponential attractor for the generalized symmetric regularized long wave equation with damping term. Appl. Math. Mech., Engl. Ed. 26 (2005), 283–291.

    Article  MathSciNet  Google Scholar 

  31. X. Shao, G. Xue, C. Li: A conservative weighted finite difference scheme for regularized long wave equation. Appl. Math. Comput. 219 (2013), 9202–9209.

    MathSciNet  MATH  Google Scholar 

  32. A. A. Soliman: Numerical scheme based on similarity reductions for the regularized long wave equation. Int. J. Comput. Math. 81 (2004), 1281–1288.

    Article  MathSciNet  Google Scholar 

  33. B. Wang, T. Sun, D. Liang: The conservative and fourth-order compact finite difference schemes for regularized long wave equation. J. Comput. Appl. Math. 356 (2019), 98–117.

    Article  MathSciNet  Google Scholar 

  34. S. Xie, S. Kim, G. Woo, S. Yi: A numerical method for the generalized regularized long wave equation using a reproducing kernel function. SIAM J. Sci. Comput. 30 (2008), 2263–2285.

    Article  MathSciNet  Google Scholar 

  35. J. Yan, M.-C. Lai, Z. Li, Z. Zhang: New conservative finite volume element schemes for the modified regularized long wave equation. Adv. Appl. Math. Mech. 9 (2017), 250–271.

    Article  MathSciNet  Google Scholar 

  36. L. Zhang: A finite difference scheme for generalized regularized long-wave equation. Appl. Math. Comput. 168 (2005), 962–972.

    MathSciNet  MATH  Google Scholar 

  37. K. Zheng, J. Hu: High-order conservative Crank-Nicolson scheme for regularized long wave equation. Adv. Difference Equ. 2013 (2013), Article ID 287, 12 pages.

  38. Y. Zhou: Applications of Discrete Functional Analysis to the Finite Difference Method. International Academic Publishers, Beijing, 1991.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, China

    Yuesheng Luo & Xiaole Li

  2. Shenyang LiaoHai Equipment CO., LTD., Shenyang, 110003, China

    Ruixue Xing

Authors
  1. Yuesheng Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruixue Xing
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaole Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaole Li.

Additional information

This research was supported by the National Natural Science Foundation of China (No. 11801116), the Fundamental Research Funds for the Central Universities, and Shandong Province Natural Science Foundation (ZR2019BA018).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, Y., Xing, R. & Li, X. A new energy conservative scheme for regularized long wave equation. Appl Math 66, 745–765 (2021). https://doi.org/10.21136/AM.2021.0066-20

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/AM.2021.0066-20

Keywords

MSC 2020

Search

Navigation