Skip to main content
SpringerLink
Log in

A High-Order Compact Finite Difference Scheme for the Fractional Sub-diffusion Equation

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

Abstract

Based on the weighted and shifted Grünwald operator, a new high-order compact finite difference scheme is derived for the fractional sub-diffusion equation. It is proved that the difference scheme is unconditionally stable and convergent in \(L_{\infty }\)-norm by the energy method. The convergence order is \(O(\tau ^3+h^4)\), where \(\tau \) is the temporal step size and \(h\) is the spatial step size. Although the unconditional stability and convergence of the difference scheme are obtained for all \(\alpha \in (0, \alpha ^{*}],\) where \(\alpha ^{*}=0.9569347,\) some numerical experiments show that they are valid for all \(\alpha \in (0, 1).\) Finally, some numerical examples are given to confirm 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.

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

Similar content being viewed by others

References

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

    MATH  Google Scholar 

  2. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  3. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Boston (2006)

    MATH  Google Scholar 

  4. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  5. Anastassiou, G.A.: Advances on Fractional Inequalities. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  6. Das, S.: Functional Fractional Calculus. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  7. Tomovski, Ž., Sandev, T.: Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions. Nonlinear Dyn. 71, 671–683 (2013)

    Article  MathSciNet  Google Scholar 

  8. Luchko, Y., Mainardi, F.: Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation. Cent. Eur. J. Phys. 11, 666–675 (2013)

    Google Scholar 

  9. 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  MathSciNet  MATH  Google Scholar 

  10. 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  MathSciNet  MATH  Google Scholar 

  11. Zhuang, P., Liu, F., Anh, V., Turner, I.: Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J. Appl. Math. 74, 645–667 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao, G., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, Y., 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  MathSciNet  MATH  Google Scholar 

  15. Yuste, S.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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  MathSciNet  MATH  Google Scholar 

  17. Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cui, M.: Compact alternating direction implict method for two-diemnsional time fractional diffusion equation. J. Comput. Phys. 231, 2621–2633 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lin, X., 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. Cao, J., Xu, C.: A high order schema for the numerical solution of the fractional ordinary differential equations. J. Comput. Phys. 238, 154–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gao, G., Sun, Z.Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)

    Article  MathSciNet  Google Scholar 

  24. Li, C.P., Ding, H.F.: Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Model. 38, 3802–3821 (2014)

    Article  MathSciNet  Google Scholar 

  25. Tian, W., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. (to appear). arXiv:1201.5949[math.NA]

  26. Li, C., Deng, W.H.: Second order WSGD operators II: a new family of difference schemes for space fractional advection diffusion equation. arXiv:1310.7671v1[math.NA] 29 Oct 2013

  27. M.H. Chen, W.H. Deng: WSLD operators: a class of fourth order difference approximations for space Riemann-Liouville derivative, arXiv preprint. arXiv:1304.7425,2013-arxiv.org

  28. Chen, M.H., Deng, W.H.: WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative. arXiv:1306.5900v1[math.NA] 25 Jun 2013

  29. Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional subdiffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1–15 (2014)

    Article  MathSciNet  Google Scholar 

  30. Zorich, V.A.: Mathematical Analysis II. Springer, Berlin (2004)

    MATH  Google Scholar 

  31. Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liao, H.L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Partial Differ. Equ. 26, 37–60 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2D edn. Science Press, Beijing (2012). (in Chinese)

    Google Scholar 

  34. Chan, R., Jin, X.: An Introduction to Iterative Toeplitz Solvers. SIAM, Philadelphia (2007)

    Book  MATH  Google Scholar 

  35. Gautschi, W.: Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38, 77–81 (1959)

    MathSciNet  MATH  Google Scholar 

  36. Dimitrov, Y.: Numerical approximations for fractional differential equations. arXiv:1311.3935v1

Download references

Acknowledgments

We would like to express our gratitude to the editor and the reviewers for their many valuable comments and suggestions.

Author information

Authors and Affiliations

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

    Cui-cui Ji & Zhi-zhong Sun

Authors
  1. Cui-cui Ji
    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 Cui-cui Ji.

Additional information

The research is supported by National Natural Science Foundation of China (No. 11271068).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ji, Cc., Sun, Zz. A High-Order Compact Finite Difference Scheme for the Fractional Sub-diffusion Equation. J Sci Comput 64, 959–985 (2015). https://doi.org/10.1007/s10915-014-9956-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9956-4

Keywords

Search

Navigation