Skip to main content
SpringerLink
Log in

The efficient alternating direction implicit Galerkin method for the nonlocal diffusion-wave equation in three dimensions

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This work formulates an alternating direction implicit (ADI) Galerkin scheme for the nonlocal diffusion-wave equation in three-dimensional space. The L1 discretization formula is used to approximate the fractional Caputo derivative. A fully discrete Galerkin scheme is obtained via spatial discretization based on the finite element method. Then, an ADI algorithm is designed to reduce the computational cost. The stability and convergence analyses are derived via the energy method. Numerical experiments demonstrate the theoretical results.

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. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  2. Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  3. Yi, L., Guo, B.: An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations. Sci. China (Mathematics) 57, 2285–2300 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Yi, L., Guo, B.: An \(h\)-\(p\) version of the continuous petrov-Galerkin finite element method for volterra integro-differential equations with smooth and nonsmooth kernels. SIAM J. Numer. Anal. 53, 2677–2704 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bouchaud, J., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)

    Article  MathSciNet  Google Scholar 

  6. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gorenflo, R., Mainardi, F., Vivoli, A.: Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Solitons Fractals 34, 87–103 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wyss, W.: Fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Schneider, W., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, 23–28 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129–143 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhuang, P., Liu, F.: Implicit difference approximation for the two-dimensional space-time fractional diffusion equation. J. Appl. Math. Comput. 25, 269–282 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, N., Deng, W., Wu, Y.: Finite difference/element method for a two-dimensional modified fractional diffusion equation. Adv. Appl. Math. Mech. 4, 496–518 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, C., Liu, F.: A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation. J. Appl. Math. Comput. 30, 219–236 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ren, J., Sun, Z.: Efficient Numerical solution of the multi-term time fractional diffusion-dave equation. East Asian J. Appl. Math. 5, 1–28 (2015)

    Article  MathSciNet  Google Scholar 

  16. Sun, H., Sun, Z.: A fast temporal second-order compact ADI difference scheme for the 2D multi-term fractional wave equation. Numer. Algor. 86, 761–797 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Qiao, L., Qiu, W., Xu, D.: A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem. Comput. Math. Appl. 102, 137–145 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Qiu, W., Xu, D., Chen, H., Guo, J.: An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile-immobile equation in two dimensions. Comput. Math. Appl. 80, 3156–3172 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Qiao, L., Xu, D.: Orthogonal spline collocation scheme for the multi-term time-fractional diffusion equation. Int. J. Comput. Math. 95, 1478–1493 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yang, X., Zhang, H., Xu, D.: Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 256, 824–837 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lin, Y., Xu, C., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197–221 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Yang, Y., Yan, Y., Neville, J.: Some time stepping methods for fractional diffusion problems with nonsmooth data. Comput. Methods. Appl. Math. 18, 129–146 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yin, B., Liu, Y., Li, H., Zeng, F.: A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equations. Appl. Numer. Math. 165, 56–82 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, L., Xu, D., Luo, M.: Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation. J. Comput. Phys. 255, 471–485 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, B., Luo, H., Xie, X.: Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data. SIAM J. Numer. Anal. 57, 779–798 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Douglas, J., Dupont, T.: Alternating direction Galerkin methods on rectangles. In: Hubbard, B. (ed.) Numerical Part Difference Equation II, pp. 133–214. Academic Press, New York (1971)

    Google Scholar 

  28. Jiang, H., Xu, D., Qiu, W., Zhou, J.: An ADI compact difference scheme for the two-dimensional semilinear time-fractional mobile-immobile equation. Comput. Appl. Math. 39, 1–17 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, X., Qiu, W., Zhang, H., Tang, L.: An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation. Comput. Math. Appl., (2021), Accepted

  30. Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dendy, J.: Ananalysis of some Galerkin schemes for the solution of nonlinear time dependent problems. SIAM J. Numer. Anal. 12, 541–565 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  32. Fernandes, R., Fairweather, G.: Analternating direction Galerkin method for a class of second-order hyperbolic equations in two space variables. SIAM J. Numer. Anal. 28, 1265–1281 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  33. Solomon, T., Weeks, E., Swinney, H.: Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow. Phys. Rev. Lett. 71, 3975–3979 (1993)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Qiong Huang & Wenlin Qiu

  2. School of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, People’s Republic of China

    Ren-jun Qi

Authors
  1. Qiong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ren-jun Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenlin Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenlin Qiu.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Q., Qi, Rj. & Qiu, W. The efficient alternating direction implicit Galerkin method for the nonlocal diffusion-wave equation in three dimensions. J. Appl. Math. Comput. 68, 3067–3087 (2022). https://doi.org/10.1007/s12190-021-01652-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01652-4

Keywords

Mathematics Subject Classification

Search

Navigation