Journal of Scientific Computing

, Volume 75, Issue 2, pp 1102–1127 | Cite as

Stability and Convergence Analysis of Finite Difference Schemes for Time-Dependent Space-Fractional Diffusion Equations with Variable Diffusion Coefficients

  • Xue-lei Lin
  • Michael K. Ng
  • Hai-Wei Sun


In this paper, we study and analyze Crank–Nicolson temporal discretization with high-order spatial difference schemes for time-dependent Riesz space-fractional diffusion equations with variable diffusion coefficients. To the best of our knowledge, there is no stability and convergence analysis for temporally 2nd-order or spatially jth-order (\(j\ge 3\)) difference schemes for such equations with variable coefficients. We prove under mild assumptions on diffusion coefficients and spatial discretization schemes that the resulting discretized systems are unconditionally stable and convergent with respect to discrete \(\ell ^2\)-norm. We further show that several spatial difference schemes with jth-order (\(j=1,2,3,4\)) truncation error satisfy the assumptions required in our analysis. As a result, we obtain a series of temporally 2nd-order and spatially jth-order (\(j=1,2,3,4\)) unconditionally stable difference schemes for solving time-dependent Riesz space-fractional diffusion equations with variable coefficients. Numerical results are presented to illustrate our theoretical results.


Time-dependent space-fractional diffusion equation Variable diffusion coefficients High-order finite difference schemes Stability Convergence 

Mathematics Subject Classification

26A33 35R11 65M06 65M12 


  1. 1.
    Agrawal, O.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 29, 145–155 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benson, D., Wheatcraft, S., Meerschaert, M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)CrossRefGoogle Scholar
  3. 3.
    Bouchaud, J., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Çelik, C., Duman, M.: Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, M., Deng, W., Wu, Y.: Superlinearly convergent algorithms for the two-dimensional space time Caputo Riesz fractional diffusion equation. Appl. Numer. Math. 70, 22–41 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    del Castillo-Negrete, D., Carreras, B., Lynch, V.: Fractional diffusion in Plasma turbulence. Phys. Plasmas 11, 3854–3864 (2004)CrossRefGoogle Scholar
  8. 8.
    Ding, H., Li, C.: High-order numerical algorithms for Riesz derivatives via constructing new generating functions. J. Sci. Comput. 71, 759–784 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hao, Z., Sun, Z., Cao, W.: A fourth-order approximation of fractional derivatives with its applications. J. Comput. Phys. 281, 787–805 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)zbMATHGoogle Scholar
  11. 11.
    Lei, S., Huang, Y.: Fast algorithms for high-order numerical methods for space-fractional diffusion equations. Int. J. Comput. Math. 94, 1062–1078 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lin, X., Ng, M., Sun, H.: A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations. J. Comput. Phys. 336, 69–86 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, Q., Liu, F., Gu, Y., Zhuang, P., Chen, J., Turner, I.: A meshless method based on point interpolation method (PIM) for the space fractional diffusion equation. Appl. Math. Comput. 256, 930–938 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equation. J. Comput. Appl. Math. 172, 65–77 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Meerschaert, M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ng, M.: Iterative Methods for Toeplitz Systems. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  18. 18.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  19. 19.
    Solomon, T., Weeks, E., Swinney, H.: Observation of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow. Phys. Rev. 71, 3975–3979 (1993)Google Scholar
  20. 20.
    Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann Liouville derivative. Appl. Numer. Math. 90, 22–37 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comp. 84, 1703–1727 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, H., Wang, K., Sircar, T.: A direct \(O(N\log ^2 N)\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yang, Z., Yuan, Z., Nie, Y., Wang, J., Zhu, X., Liu, F.: Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains. J. Comput. Phys. 330, 863–883 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, Y., Ding, H.: High-order algorithm for the two-dimension Riesz space-fractional diffusion equation. Int. J. Comput. Math. 94, 2063–2073 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong
  2. 2.Department of MathematicsUniversity of MacauMacaoPeople’s Republic of China

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