Skip to main content
Log in

Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation

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

Abstract

In this paper, we consider a space-time Riesz–Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 ∈ (0,1) and β 2 ∈ (1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. Burnecki, K., Janczura, J., Magdziarz, M., Weron, A.: Can one see a competition between subdiffusion and Lévy flights? A case of geometric-stable noise. Acta Phys. Pol., B 39, 1043–1054 (2008)

    Google Scholar 

  2. Chen, C., Liu, F., Turner, I., Anh, V.: Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. Numer. Algorithms 54, 1–21 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ciesielski, M., Leszczynski, J.: Numerical solutions to boundary value problem for anomalous diffusion equation with Riesz–Feller fractional operator. J. Theor. Appl. Mech. 44, 393–403 (2006)

    Google Scholar 

  4. Del-Castillo-Negrete, D.: Fractional diffusion models of nonlocal transport. Phys. Plasmas 13, 082308 (2006)

    Article  MathSciNet  Google Scholar 

  5. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractioal differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional adams method. Numer. Algorithms 36, 31–52 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Diethelm, K., Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16, 231–253 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Deng, W.: Short memory principle and a predictor-corrector approach for fractional differential equations. J. Comput. Appl. Math. 206, 174–188 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ford, N.J., Simpson, A.C.: The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 26, 336–346 (2001)

    MathSciNet  Google Scholar 

  10. Fulger, D., Scalas, E., Germano, G.: Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys. Rev., E 77, 021122 (2008)

    Article  Google Scholar 

  11. Gorenflo, R., Mainardi, F.: Approximation of Lévy–Feller diffusion by random walk. J. Anal. Appl. (ZAA) 18, 231–246 (1999)

    MATH  MathSciNet  Google Scholar 

  12. Gorenflo, R., Vivoli, A.: Fully discrete random walks for space-time fractional diffusion equations. Signal Process. 83, 2411–2420 (2003)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, F., Anh, V., Turner, I., Zhuang, P.: Numerical simulation for solute transport in fractal porous media. Australian and New Zealand Industrial and Applied Mathematics Journal 45(E), 461–473 (2004)

    MathSciNet  Google Scholar 

  16. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Magdziarz, M., Weron, A.: Fractional Fokker–Planck dynamics: stochastic representation and computer simulation. Phys. Rev., E 75, 016708 (2007)

    Article  Google Scholar 

  18. Magdziarz, M., Weron, A.: Competition between subdiffusion and Lévy fights: a Monte Carlo approach. Phys. Rev., E 75, 056702 (2007)

    Article  Google Scholar 

  19. Meerschaert, M.M., Scheffler, H., Tadjeran, C.: Finite difference methods for two dimensional fractional dispersion equation. J. Comput. Phys. 211, 249–261 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. 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 

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

    MATH  Google Scholar 

  24. Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2. J. Comput. Appl. Math. 193, 243–268 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. 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 

  26. Shen, S., Liu, F., Anh, V.: Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order. J. Appl. Math. Comput. 28 , 147–164 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220, 813–823 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zhuang, P., Liu, F.: Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22, 87–99 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fawang Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, S., Liu, F. & Anh, V. Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation. Numer Algor 56, 383–403 (2011). https://doi.org/10.1007/s11075-010-9393-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-010-9393-x

Keywords

Navigation