Skip to main content
SpringerLink
Log in

Unconditional Energy Dissipation and Error Estimates of the SAV Fourier Spectral Method for Nonlinear Fractional Generalized Wave Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we consider a second-order scalar auxiliary variable Fourier spectral method to solve the nonlinear fractional generalized wave equation. Unconditional energy conservation or dissipation properties of the fully discrete scheme are first established. Next, we utilize the temporal-spatial error splitting argument to obtain unconditional optimal error estimate of the fully discrete scheme, which overcomes time-step restrictions caused by strongly nonlinear system, or the restrictions that the nonlinear term needs to satisfy the assumption of global Lipschitz condition in all previous works for fractional undamped or damped wave equations. Finally, some numerical experiments are presented to confirm our 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

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

Similar content being viewed by others

References

  1. Bratsos, A.G.: On the numerical solution of the Klein–Gordon equation. Numer. Methods Partial Differ. Equ. 25, 939–951 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Rashidinia, J., Ghasemi, M., Jalilian, R.: Numerical solution of the nonlinear Klein–Gordon equation. J. Comput. Appl. Math. 233, 1866–1878 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Deng, D., Zhang, C.: Analysis and application of a compact multistep ADI solver for a class of nonlinear viscous wave equations. Appl. Math. Model. 39, 1033–1049 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Campa, A., Dauxois, T., Ruffo, S.: Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Rep. 480, 57–159 (2009)

    Article  MathSciNet  Google Scholar 

  5. Shomberg, J.L.: Well-posedness of semilinear strongly damped wave equations with fractional diffusion operators and \(C^0\) potentials on arbitrary bounded domains. Rocky Mt. J. Math. 49, 1307–1334 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, S., Jiang, X., Liu, F., Turner, I.: High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation. J. Comput. Appl. Math. 278, 119–129 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ran, M., Zhang, C.: Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions. Comput. Math. Appl. 71, 1151–1162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bu, W., Tang, Y., Wu, Y., Yang, J.: Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J. Comput. Phys. 293, 264–279 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, M., Gu, X.-M., Huang, C., Fei, M., Zhang, G.: A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations. J. Comput. Phys. 358, 256–282 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, M., Huang, C., Zhao, Y.: Fast conservative numerical algorithm for the coupled fractional Klein–Gordon–Schrödinger equation. Numer. Algorithms 84, 1081–1119 (2019)

    Article  MATH  Google Scholar 

  11. Wang, Y., Mei, L.: A conservative spectral Galerkin method for the coupled nonlinear space-fractional Schrödinger equations. Int. J. Comput. Math. 96, 2387–2410 (2019)

    Article  MathSciNet  Google Scholar 

  12. Wang, P., Huang, C.: Structure-preserving numerical methods for the fractional Schrödinger equation. Appl. Numer. Math. 129, 137–158 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, P., Huang, C.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, J., Xiao, A.: Conservative Fourier spectral method and numerical investigation of space fractional Klein–Gordon–Schrödinger equations. Appl. Math. Comput. 350, 348–365 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Wang, N., Huang, C.: An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations. Comput. Math. Appl. 75, 2223–2242 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, H., Jiang, X., Wang, C., Chen, S.: Crank–Nicolson Fourier spectral methods for the space fractional nonlinear Schrödinger equation and its parameter estimation. Int. J. Comput. Math. 96, 238–263 (2019)

    Article  MathSciNet  Google Scholar 

  17. Xing, Z., Wen, L.: A conservative difference scheme for the Riesz space-fractional sine-Gordon equation. Adv. Differ. Equ. 238, 22 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Fu, Y., Cai, W., Wang, Y.: An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation. Appl. Math. Lett. 102, 106–123 (2020)

    Article  MathSciNet  Google Scholar 

  19. Macías-Díaz, J.E.: An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions. Commun. Nonlinear Sci. Numer. Simul. 59, 67–87 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Macías-Díaz, J.E., Hendy, A.S., De Staelen, R.H.: A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives. Comput. Phys. Commun. 224, 98–107 (2018)

    Article  MathSciNet  Google Scholar 

  21. Macías-Díaz, J.E.: A numerically efficient dissipation-preserving implicit method for a nonlinear multidimensional fractional wave equation. J. Sci. Comput. 77, 1–26 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Macías-Díaz, J.E., Hendy, A.S., De Staelen, R.H.: A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations. Appl. Math. Comput. 325, 1–14 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Xie, J., Zhang, Z.: An effective dissipation-preserving fourth-order difference solver for fractional-in-space nonlinear wave equations. J. Sci. Comput. 79, 1753–1776 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xie, J., Zhang, Z., Liang, D.: A new fourth-order energy dissipative difference method for high-dimensional nonlinear fractional generalized wave equations. Commun. Nonlinear Sci. Numer. Simul. 78, 104850 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Matsuo, T., Furihata, D.: Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171, 425–447 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang, X., Zhao, J., Wang, Q., Shen, J.: Numerical approximations for a three-component Cahn–Hilliard phase-field model based on the invariant energy quadratization method. Math. Models Methods Appl. Sci. 27, 1993–2030 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, Y.-W., Wu, X.: General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs. J. Comput. Phys. 301, 141–166 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gong, Y., Cai, J., Wang, Y.: Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs. J. Comput. Phys. 279, 80–102 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jiang, C., Cai, W., Wang, Y.: A linearly implicit and local energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach. J. Sci. Comput. 80, 1629–1655 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61, 474–506 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, X., Shen, J.: Stability and Error Estimates of the SAV Fourier-Spectral Method for the Phase Field Crystal Equation. arXiv (2019)

  33. Jiang, C., Gong, Y., Cai, W., Wang, Y.: A linearly implicit structure-preserving scheme for the Camassa–Holm equation based on multiple scalar auxiliary variables approach. J. Sci. Comput. 80, 1629–1655 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, X., Shen, J., Rui, H.: Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Math. Comput. 88, 2047–2068 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I., Anh, V.: A Crank–Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction–diffusion equation. SIAM J. Numer. Anal. 52, 2599–2622 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, N., Fei, M., Huang, C., Zhang, G., Li, M.: Dissipation-preserving Galerkin–Legendre spectral methods for two-dimensional fractional nonlinear wave equations. Comput. Math. Appl. 80, 617–635 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959–1977 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, D., Wang, J.: Unconditionally optimal error analysis of Crank–Nicolson Galerkin FEMs for a strongly nonlinear parabolic system. J. Sci. Comput. 72, 892–915 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Si, Z., Wang, J., Sun, W.: Unconditional stability and error estimates of modified characteristics FEMs for the Navier–Stokes equations. Numer. Math. 134, 139–161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhang, H., Jiang, X., Zeng, F., Karniadakis, G.E.: A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction–diffusion equations. J. Comput. Phys. 405, 109–141 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ainsworth, M., Mao, Z.: Analysis and approximation of a fractional Cahn–Hilliard equation. SIAM J. Numer. Anal. 55, 1689–1718 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Adams, R.A., Fournier, J.F.: Sobolev Spaces, vol. 140, pp. C713–C734. Elsevier, Amsterdam (2003)

    Google Scholar 

  43. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous reviewer for excellent suggestions that helped improve this paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Nan Wang & Meng Li

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Chengming Huang

  3. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Chengming Huang

Authors
  1. Nan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Meng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chengming Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Meng Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by NSF of China (12001499, 11771163, 11801527, 12011530058), China Postdoctoral Science Foundation (2019M662506, 2018M632791).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, N., Li, M. & Huang, C. Unconditional Energy Dissipation and Error Estimates of the SAV Fourier Spectral Method for Nonlinear Fractional Generalized Wave Equation. J Sci Comput 88, 19 (2021). https://doi.org/10.1007/s10915-021-01534-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01534-8

Keywords

Search

Navigation