Skip to main content

Advertisement

SpringerLink
Log in

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

  • Original Paper
  • Published:
Communications on Applied Mathematics and Computation Aims and scope Submit manuscript

Abstract

In this paper, two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation. Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense, the new schemes are proved to perfectly preserve the total energy in the discrete sense. By using the standard energy method and the cut-off function technique, the optimal error estimates of the numerical solutions are established, and the convergence rates are of \(O(h^4+\tau ^2)\) with mesh-size h and time-step \(\tau \). In order to improve the computational efficiency, an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step. The convergence of the iterative algorithm is also rigorously analyzed. Several numerical results are carried out to test the error estimates and conservative properties.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Achouri, T.: Conservative finite difference scheme for the nonlinear fourth-order wave equation. Appl. Math. Comput. 359, 121–131 (2019)

    Article  MathSciNet  Google Scholar 

  2. Akrivis, G.D.: Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal. 13, 115–124 (1993)

    Article  MathSciNet  Google Scholar 

  3. Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math. 59, 31–53 (1991)

    Article  MathSciNet  Google Scholar 

  4. Bao, W., Cai, Y.: Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation. Math. Comput. 82, 99–128 (2013)

    Article  MathSciNet  Google Scholar 

  5. Berger, K.M., Milewski, P.A.: Simulation of wave interactions and turbulence in one-dimensional water waves. SIAM J. Appl. Math. 63, 1121–1140 (2003)

    Article  MathSciNet  Google Scholar 

  6. Berloff, N.G., Howard, L.N.: Nonlinear wave interactions in nonlinear nonintegrable systems. Stud. Appl. Math. 100, 195–213 (1998)

    Article  MathSciNet  Google Scholar 

  7. Bernier, J., Feola, R., Grébert, B., Iandoli, F.: Long-time existence for semi-linear beam equations on irrational tori. J. Dyn. Differ. Equ. 143, 1–36 (2021)

    MathSciNet  MATH  Google Scholar 

  8. Bretherton, F.P.: Resonant interaction between waves: the case of discrete oscillations. J. Fluid Mech. 20, 457–479 (1964)

    Article  MathSciNet  Google Scholar 

  9. Cai, Y., Yuan, Y.J.: Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime. Math. Comput. 87(311), 1191–1225 (2018)

    Article  MathSciNet  Google Scholar 

  10. Gupta, C.P.: Existence and uniqueness results for the bending of an elastic beam equation at resonance. J. Math. Anal. Appl. 135, 208–225 (1988)

    Article  MathSciNet  Google Scholar 

  11. Haddadpour, H.: An exact solution for variable coefficients fourth-order wave equation using the Adomian method. Math. Comput. Model. 44, 1144–1152 (2006)

    Article  MathSciNet  Google Scholar 

  12. Han, S.M., Benaroya, H., Wei, T.: Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib. 225, 935–988 (1999)

    Article  Google Scholar 

  13. Levandosky, S.: Decay estimates for fourth order wave equations. J. Differ. Equ. 143, 360–413 (1998)

    Article  MathSciNet  Google Scholar 

  14. Levandosky, S.: Stability and instability of fourth-order solitary waves. J. Dyn. Differ. Equ. 10, 151–188 (1998)

    Article  MathSciNet  Google Scholar 

  15. Levandosky, S., Strauss, W.A.: Time decay for the nonlinear beam equation. Methods Appl. Anal. 7, 479–488 (2000)

    Article  MathSciNet  Google Scholar 

  16. Levine, H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form \(pu_{tt}=-au+f(u)\). Trans. Am. Math. Soc. 192, 1–21 (1974)

    MATH  Google Scholar 

  17. Levine, H.A., Park, S.R., Serrin, J.: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J. Math. Anal. Appl. 228, 181–205 (1998)

    Article  MathSciNet  Google Scholar 

  18. Li, B., Fairweather, G., Bialecki, B.: Discrete-time orthogonal spline collocation methods for vibration problems. SIAM J. Numer. Anal. 39, 2045–2065 (2002)

    Article  MathSciNet  Google Scholar 

  19. Li, J., Sun, Z., Zhao, X.: A three level linearized compact difference scheme for the Cahn-Hilliard equation. Sci. China Math. 55(04), 805–826 (2012)

    Article  MathSciNet  Google Scholar 

  20. Lin, J.E.: Local time decay for a nonlinear beam equation. Methods Appl. Anal. 11, 065–068 (2004)

    Article  MathSciNet  Google Scholar 

  21. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Annales de l'Institut Henri Poincaré C, Analyse non linéaire 1, 109–145 (1984)

    Article  Google Scholar 

  22. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)

    MATH  Google Scholar 

  23. Mattsson, K., Stiernström, V.: High-fidelity numerical simulation of the dynamic beam equation. J. Comput. Phys. 286, 194–213 (2015)

    Article  MathSciNet  Google Scholar 

  24. Pausader, B.: Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations. J. Differ. Equ. 241, 237–278 (2007)

    Article  MathSciNet  Google Scholar 

  25. Peletier, L., Troy, W.C.: Spatial Patterns: Higher Order Models in Physics and Mechanics. In: Brezis, H. (ed) Progress in Nonlinear Differential Equations and Their Applications, vol. 45. Birkhäuser, Basel (2001)

    Book  Google Scholar 

  26. Takeda, H., Yoshikawa, S.: On the initial value problem of the semilinear beam equation with weak damping II: asymptotic profiles. J. Differ. Equ. 253(11), 3061–3080 (2012)

    Article  MathSciNet  Google Scholar 

  27. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. In: Bank, R., Graham, R.L., Stoer, J., Varga, R., Yserentant, H. (eds) Springer Series in Computational Mathematics, vol. 25. Springer-Verlag, Berlin (1997)

  28. Wang, T., Guo, B., Xu, Q.: Fourth-order compact and energy conservative difference scheme for the nonlinear Schrödinger equation in two dimensions. J. Comput. Phys. 243, 382–399 (2013)

    Article  MathSciNet  Google Scholar 

  29. Ye, Y.: Global existence and blow-up of solutions for higher-order viscoelastic wave equation with a nonlinear source term. Nonlinear Anal. Theor. 112, 129–146 (2015)

    Article  MathSciNet  Google Scholar 

  30. Zhang, G.: Two conservative and linearly-implicit compact difference schemes for the nonlinear fourth-order wave equation. Appl. Math. Comput. 401, 001–013 (2021)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 11571181, and the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20171454.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, China

    Xiaoyi Liu, Tingchun Wang, Shilong Jin & Qiaoqiao Xu

Authors
  1. Xiaoyi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tingchun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shilong Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qiaoqiao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tingchun Wang.

Ethics declarations

Conflict of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Wang, T., Jin, S. et al. Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation. Commun. Appl. Math. Comput. 4, 1509–1530 (2022). https://doi.org/10.1007/s42967-022-00193-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42967-022-00193-2

Keywords

Mathematics Subject Classification

Search

Navigation