Skip to main content
SpringerLink
Log in

Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion Equation

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

Abstract

In this paper, a Crank–Nicolson-type compact ADI scheme is proposed for solving two-dimensional fractional subdiffusion equation. The unique solvability, unconditional stability and convergence of the scheme are proved rigorously. Two error estimates are presented. One is \(\mathcal{O }(\tau ^{\min \{2-\frac{\gamma }{2},\,2\gamma \}}+h_1^4+h^4_2)\) in standard \(H^1\) norm, where \(\tau \) is the temporal grid size and \(h_1,h_2\) are spatial grid sizes; the other is \(\mathcal{O }(\tau ^{2\gamma }+h_1^4+h^4_2)\) in \(H^1_{\gamma }\) norm, a generalized norm which is associated with the Riemann–Liouville fractional integral operator. Numerical results are presented to support 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. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic, New york (1974)

    MATH  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)

    MATH  Google Scholar 

  3. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29(1–4), 129–143 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von-Neumann type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, C.M., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing subdiffusion. J. Comput. Phys. 227, 886–897 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen, C.M., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32, 1740–1760 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gao, G.H., Sun, Z.Z.: A compact difference scheme for the fractional subdiffusion equations. J. Comput. Phys. 230, 586–595 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Deng, W.: Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys. 227, 1510–1512 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  18. Zhao, X., Sun, Z.Z.: A box-type scheme for fractional subdiffusion equation with Neumann boundary conditions. J. Comput. Phys. 230, 6061–6074 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lin, Y., Li, X., Xu, C.: Finite difference/spectral approximations for the fractional cable equation. Math. Comput. 80, 1369–1396 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhang, Y.N., Sun, Z.Z., Wu, H.W.: Error estimates of Crank–Nicolson type difference schemes for the subdiffusion equation. SIAM J. Numer. Anal. 49, 2302–2322 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229, 6613–6622 (2010)

    Google Scholar 

  22. Zhang, Y.N., Sun, Z.Z.: Alternating direction implicit schemes for the two-dimensional fractional subdiffusion equation. J. Comput. Phys. 230, 8713–8728 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Cui, M.R.: Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J. Comput. Phys. 231, 2621–2633 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit schemes for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Samarskii, A.A., Andreev, V.B.: Difference Methods for Elliptic Equation. Nauka, Moscow (1976)

    Google Scholar 

  26. Sun, Z.Z.: The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations. Science Press, Beijing (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, People’s Republic of China

    Ya-nan Zhang

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

    Zhi-zhong Sun

Authors
  1. Ya-nan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-zhong Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ya-nan Zhang.

Additional information

The research is supported by National Natural Science Foundation of China (Contract Grant number 11271068) and China postdoctoral Science Foundation (2013M530265).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Yn., Sun, Zz. Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion Equation. J Sci Comput 59, 104–128 (2014). https://doi.org/10.1007/s10915-013-9756-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-013-9756-2

Keywords

Search

Navigation