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On the L\(^{\infty }\) convergence of a novel fourth-order compact and conservative difference scheme for the generalized Rosenau-KdV-RLW equation

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Abstract

In this work, a novel compact difference scheme for the generalized Rosenau-KdV-RLW equation is investigated, which preserves some conservative properties of the original equation. The proposed numerical scheme is three-level and nonlinear implicit, which adopts a novel approximation \(\displaystyle \frac{1}{4}(U^{n+1}_{j}+2U^{n}_{j}+U^{n-1}_{j})\) of \(u(x_{j},t_{n})\) in the temporal direction and a direct compact discretization method for the higher-order derivatives \(u_{xxx}\) and \(u_{xxxx}\) in the spatial direction. The discrete conservation laws are given, and a priori estimates of the difference solution in maximum norm are obtained by applying mathematical induction. The solvability of the proposed numerical scheme is proved by the well-known Brouwer’s fixed point theorem. The convergence with \(O(\tau ^{2}+h^{4})\) convergence order and stability of the novel difference scheme in \(L^{\infty }\) norm are proved. A linearized iterative algorithm for solving nonlinear algebraic systems generated by the compact scheme is proposed and its convergence is proved. Some numerical experimental results validate the theoretical results and demonstrate that the proposed compact scheme is efficient and reliable.

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References

  1. Korteweg, D.J., De Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Dublin Philosoph. Mag. J. Sci. 39(240), 422–443 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  2. Peregrine, D.H.: Long waves on a beach. J. Fluid Mech. 27(4), 815–827 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Seyler, C.E., Fenstermacher, D.L.: A symmetric regularized-long-wave equation. Phys. Fluids. 27(1), 4–7 (1984)

    Article  MATH  Google Scholar 

  4. Rosenau, P.: A quasi-continuous description of a nonlinear transmission line. Phys. Scr. 34(6B), 827–829 (1986)

    Article  Google Scholar 

  5. Razborova, P., Kara, A.H., Biswas, A.: Additional conservation laws for Rosenau-KdV-RLW equation with power law nonlinearity by lie symmetry. Nonlinear Dyn. 79(1), 743–748 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wongsaijai, B., Poochinapan, K.: Optimal decay rates of the dissipative shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation with power of nonlinearity. Appl. Math. Comput. 405(126202) (2021)

  7. Atouani, N., Ouali, Y., Omrani, K.: Mixed finite element methods for the Rosenau equation. J. Appl. Math. Comput. 57(1), 393–420 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Abbaszadeh, M., Dehghan, M.: The interpolating element-free Galerkin method for solving Korteweg-de Vries-Rosenau-regularized long-wave equation with error analysis. Nonlinear Dyn. 96(2), 1345–1365 (2019)

    Article  MATH  Google Scholar 

  9. Ji, B., Zhang, L., Sun, Q.: A dissipative finite difference Fourier pseudospectral method for the symmetric regularized long wave equation with damping mechanism. Appl. Numer. Math. 154, 90–103 (2022)

    Article  MATH  Google Scholar 

  10. Ahmat, M., Qiu, J.: Compact ETDRK scheme for nonlinear dispersive wave equations. Comput. Appl. Math. 40(8), 1–17 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ozer, S.: Numerical solution of the Rosenau-KdV-RLW equation by operator splitting techniques based on B-spline collocation method. Numer. Methods Partial Differential Equations 35(5), 1928–1943 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Apolinar-Fernandez, A., Ramos, J.I.: Numerical solution of the generalized, dissipative KdV-RLW-Rosenau equation with a compact method. Commun. Nonlinear Sci. Numer. Simul. 60, 165–183 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, S.G.: Numerical study of a conservative weighted compact difference scheme for the symmetric regularized long wave equations. Numer. Methods Partial Differential Equations 35(1), 60–83 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, T.C., Guo, B.L.: Unconditional convergence of two conservative compact difference schemes for non-linear Schrodinger equation in one dimension. Sci. Sin. Math. 40(3), 207–233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, S.G.: Numerical analysis for fourth-order compact conservative difference scheme to solve the 3D Rosenau-RLW equation. Comput. Math. Appl. 72(9), 2388–2407 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, X., Gong, Y., Zhang, L.: Two novel classes of linear high-order structure-preserving schemes for the generalized nonlinear Schrodinger equation. Appl. Math. Lett. 104(106273) (2020)

  18. Wongsaijai, B., Poochinapan, K.: A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau-KdV equation and the Rosenau-RLW equation. Appl. Math. Comput. 245(2), 289–304 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Wang, X., Dai, W.: A three-level linear implicit conservative scheme for the Rosenau-KdV-RLW equation. J. Comput. Appl. Math. 330, 295–306 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, X., Dai, W.: A new conservative finite difference scheme for the generalized Rosenau-KdV-RLW equation. Comput. Appl. Math. 39(3), 1–19 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ghiloufi, A., Omrani, K.: New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves. Numer. Methods Partial Differential Equations. 34(2), 451–500 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, S.G., Wu, X.: L\(^{\infty }\) error bound of conservative compact difference scheme for the generalized symmetric regularized long-wave (GSRLW) equations. Comput. Appl. Math. 37(3), 2816–2836 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dimitrienko, Y.I., Li, S.G., Niu, Y.: Study on the dynamics of a nonlinear dispersion model in both 1D and 2D based on the fourth-order compact conservative difference scheme. Math. Comput. Simul. 182, 661–689 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zuo, J.M., Zhang, Y.M., Zhang, T.D., Chang, F.: A new conservative difference scheme for the general Rosenau-RLW equation. Bound. Value Probl. 2010, 1–13 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wongsaijai, B., Poochinapan, K., Disyadej, T.: A compact finite difference method for solving the general Rosenau-RLW equation. Int. J. Appl. Math. 44(4), 192–199 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Pan, X., Zhang, L.: Numerical simulation for general Rosenau-RLW equation: An average linearized conservative scheme. Math. Probl. Eng. 15(517818) (2012)

  27. Hu, J., Xu, Y., Hu, B.: Conservative linear difference scheme for Rosenau-KDV equation. Adv. Math. Phys. 2013(423718) (2013)

  28. Zheng, M., Zhou, J.: An average linear difference scheme for the generalized Rosenau-KDV equation. J. Appl. Math. 2014(202793) (2014)

  29. Luo, Y., Xu, Y., Feng, M.: Conservative difference scheme for generalized Rosenau-KDV equation. Adv. Math. Phys. 2014(986098) (2014)

  30. Browder, F.E.: Existence and uniqueness theorems for solutions of nonlinear boundary value problems. Proc. Sympos. Appl. Math. 17, 24–49 (1965)

    Article  MathSciNet  Google Scholar 

  31. Zhou, Y.: Application of Discrete Functional Analysis to the Finite Difference Method. Inter. Acad. Publishers, Beijing (1990)

  32. Li, S.G., Xu, D., Zhang, J., Sun, C.: A new three-level fourth-order compact finite difference scheme for the extended Fisher-Kolmogorov equation. Appl. Numer. Math. 178(240), 41–51 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sun, Z.Z., Zhu, Q.D.: On Tsertsvadze’s difference scheme for the Kuramoto-Tsuzuki equation. J. Comput. Appl. Math. 98(2), 289–304 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author’s work is supported partially by the Natural Science Foundation of Liaoning Province of China (No. 2022-BS-093) and the Fundamental Research Funds for the Central Universities (No. 3132022202). The third author’s work is supported partially by the “Educational Science Planning Projects of Liaoning Province of China (JG21DB065)”. The authors would like to thank the editors and anonymous reviewers for their valuable comments and suggestions, which are very helpful for improving the quality of the article.

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Authors and Affiliations

  1. School of Science, Dalian Maritime University, No. 1 Linghai Road, Dalian, 116026, China

    Shuguang Li & Kai Qu

  2. Federal Research Center Computer Science and Control (FRC CSC), Russian Academy of Sciences (RAS), Prospekt 60-letiya Oktyabrya, d.9, Moscow, 119333, Russia

    Oleg V. Kravchenko

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  1. Shuguang Li
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Correspondence to Shuguang Li.

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Li, S., Kravchenko, O.V. & Qu, K. On the L\(^{\infty }\) convergence of a novel fourth-order compact and conservative difference scheme for the generalized Rosenau-KdV-RLW equation. Numer Algor 94, 789–816 (2023). https://doi.org/10.1007/s11075-023-01520-1

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