Skip to main content
SpringerLink
Log in

Preconditioned quasi-compact boundary value methods for space-fractional diffusion equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

This paper focuses on highly efficient numerical methods for solving space-fractional diffusion equations. By combining the fourth-order quasi-compact difference scheme and boundary value methods, a class of quasi-compact boundary value methods are constructed. In order to accelerate the convergence rate of this class of methods, the Kronecker product splitting (KPS) iteration method and the preconditioned method with KPS preconditioner are proposed. A convergence criterion for the KPS iteration method is derived. A numerical experiment further illustrates the computational efficiency and accuracy of the proposed methods. Moreover, a numerical comparison with the preconditioned method with Strang-type preconditioner is given, which shows that the preconditioned method with KPS preconditioner is comparable in computational efficiency.

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

Similar content being viewed by others

References

  1. Amodio, P., Brugnano, L.: The conditioning of Toeplitz banded matrices. Math. Comput. Modelling 23, 29–42 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Bai, J., Feng, X.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16, 2492–2502 (2007)

    MathSciNet  Google Scholar 

  3. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)

    Google Scholar 

  4. Brugnano, L.: Essentially symplectic boundary value methods for linear Hamiltonian systems. J. Comput. Math. 15, 233–252 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Brugnano, L., Trigiante, D.: Tridiagonal matrices: invertibility and conditioning. Linear Algebra Appl. 166, 131–150 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Brugnano, L., Trigiante, D.: Convergence and stability of boundary value methods for ordinary differential equations. J. Comput. Appl. Math. 66, 97–109 (1996)

    MathSciNet  MATH  Google Scholar 

  7. Brugnano, L., Trigiante, D.: Solving differential equations by multistep initial and boundary value methods. Gordan and Breach, Amsterdam (1998)

    MATH  Google Scholar 

  8. Brugnano, L., Trigiante, D.: Boundary value methods: the third way between linear multistep and Runge-Kutta methods. Comput. Math. Appl. 36, 269–284 (1998)

    MathSciNet  MATH  Google Scholar 

  9. Brugnano, L., Zhang, C., Li, D.: A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schrödinger equation with wave operator. Comm. Nonlinear Sci. Numer. Simmu. 60, 33–49 (2018)

    Google Scholar 

  10. Chan, R.H., Ng, M.K., Jin, X.: Strang-type preconditioners for systems of LMF-based ODE codes. IMA J. Numer. Anal. 21, 451–462 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Chen, H., Zhang, C.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218, 2619–2630 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Chen, H., Zhang, C.: Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations. Appl. Numer. Math. 62, 141–154 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Chen, H.: A splitting preconditioner for the iterative solution of implicit Runge-Kutta and boundary value methods. BIT 54, 607–621 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Chen, H.: Kronecker product splitting preconditioners for implicit Runge-Kutta discretizations of viscous wave equations. Appl. Math. Model. 40, 4429–4440 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Chen, H.: A splitting preconditioner for implicit Runge-Kutta discretizations of a partial differential-algebraic equation. Numer. Algor. 73, 1037–1054 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Chen, H., Zhang, T., Lv, W.: Block preconditioning strategies for time-space fractional diffusion equations. Appl. Math. Comput. 337, 41–53 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Chen, M., Deng, W.: Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators. Comm. Comput. Phys. 16, 516–540 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Davis, P.: Circulant matrices. AMS Chelsea Publishing, Rhode Island (1994)

    MATH  Google Scholar 

  20. Gal, N., Weihs, D.: Experimental evidence of strong anomalous diffusion in living cells. Phys. Rev. E 81, 020903 (2010)

    Google Scholar 

  21. Ganti, V., Meerschaert, M.M., Georgiou, E.F., Viparelli, E., Parker, G.: Normal and anomalous diffusion of gravel tracer particles in rivers. J. Geophys. Res. 115, F00A07 (2010)

    Google Scholar 

  22. Gu, X., Huang, T., Zhao, X., Li, H., Li, L.: Strang-type preconditioners for solving fractional diffusion equations by boundary value methods. J. Comput. Appl. Math. 277, 73–86 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Hao, Z., Sun, Z., Cao, W.: A fourth-order approximation of fractional derivatives with its applications. J. Comput. Phys. 281, 787–805 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  25. Iavernaro, F., Mazzia, F.: Convergence and stability of multistep methods solving nonlinear initial value problems. SIAM J. Sci. Comput. 18, 270–285 (1997)

    MathSciNet  MATH  Google Scholar 

  26. Iavernaro, F., Mazzia, F.: Block-boundary value methods for the solution of ordinary differential equations. SIAM J. Sci. Comput. 21, 323–339 (1999)

    MathSciNet  MATH  Google Scholar 

  27. Lei, S., Huang, Y.: Fast algorithms for high-order numerical methods for space fractional diffusion equations. Int. J. Comput. Math. 94, 1062–1078 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Lei, S., Sun, H.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Li, C., Zhang, C.: Block boundary value methods applied to functional differential equations with piecewise continuous arguments. Appl. Numer. Math. 115, 214–224 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Li, C., Zhang, C.: The extended generalized Störmer-Cowell methods for second-order delay boundary value problems. Appl. Math. Comput. 294, 87–95 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  33. Oldham, K.B.: Fractional differential equations in electrochemistry. Adv. Engrg. Software 41, 9–12 (2010)

    MATH  Google Scholar 

  34. Pang, H., Sun, H.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231, 693–703 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  36. Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314, 749–755 (2002)

    MATH  Google Scholar 

  37. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    MathSciNet  MATH  Google Scholar 

  38. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: theory and applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  39. Scher, H., Montroll, E.W.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455–2477 (1975)

    Google Scholar 

  40. Wang, H., Du, N.: A super fast-preconditioned iterative method for steady-state space-fractional diffusion equations. J. Comput. Phys. 240, 49–57 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Wang, H., Wang, K., Sircar, T.: A direct \(\mathcal {O}(N\log _{2} {N)}\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Wang, H., Zhang, C., Zhou, Y.: A class of compact boundary value methods applied to semi-linear reaction-diffusion equations. Appl. Math. Comput. 325, 69–81 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Zhang, C., Chen, H.: Block boundary value methods for delay differential equations. Appl. Numer. Math. 60, 915–923 (2010)

    MathSciNet  MATH  Google Scholar 

  44. Zhang, C., Chen, H.: Asymptotic stability of block boundary value methods for delay differential-algebraic equations. Math. Comput. Simul. 81, 100–108 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Zhang, C., Chen, H., Wang, L.: Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations. Numer. Linear Alge. Appl. 18, 843–855 (2011)

    MathSciNet  MATH  Google Scholar 

  46. Zhang, C., Li, C.: Generalized Störmer-Cowell methods for nonlinear BVPs of second-order delay-integro-differential equations. J. Sci. Comput. 74, 1221–1240 (2018)

    MathSciNet  MATH  Google Scholar 

  47. Zhou, Y., Zhang, C., Wang, H.: The extended boundary value methods for a class of fractional differential equations with Caputo derivatives, submitted

  48. Zhou, Y., Zhang, C.: Convergence and stability of block boundary value methods applied to nonlinear fractional differential equations with Caputo derivatives. Appl. Numer. Math. 135, 367–380 (2019)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work is supported by NSFC (Grant no. 11571128).

Author information

Authors and Affiliations

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

    Yongtao Zhou & Chengjian Zhang

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

    Yongtao Zhou & Chengjian Zhang

  3. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, Florence, I-50134, Italy

    Luigi Brugnano

Authors
  1. Yongtao Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chengjian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luigi Brugnano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chengjian Zhang.

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

Zhou, Y., Zhang, C. & Brugnano, L. Preconditioned quasi-compact boundary value methods for space-fractional diffusion equations. Numer Algor 84, 633–649 (2020). https://doi.org/10.1007/s11075-019-00773-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00773-z

Keywords

Search

Navigation