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Quasi-Compact Finite Difference Schemes for Space Fractional Diffusion Equations

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Abstract

In this paper, a compact difference operator, termed CWSGD, is designed to establish the quasi-compact finite difference schemes for approximating the space fractional diffusion equations in one and two dimensions. The method improves the spatial accuracy order of the weighted and shifted Grünwald difference (WSGD) scheme (Tian et al., arXiv:1201.5949) from 2 to 3. The numerical stability and convergence with respect to the discrete L 2 norm are theoretically analyzed. Numerical examples illustrate the effectiveness of the quasi-compact schemes and confirm the theoretical estimations.

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References

  1. Barkai, E.: CTRW pathways to the fractional diffusion equation. Chem. Phys. 284, 13–27 (2002)

    Article  Google Scholar 

  2. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)

    Article  Google Scholar 

  3. Chaves, A.S.: A fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239, 13–16 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1, 167–191 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Laub, A.J.: Matrix Analysis for Scientists and Engineers. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  6. Leveque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, Philadelphia (2007)

    MATH  Google Scholar 

  7. Marchuk, G.I., Shaidurov, V.V.: Difference Methods and Their Extrapolations. Springer, New York (1983)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A, Math. Gen. 37, R161–R208 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Meerschaert, M.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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn & Bacon, Needham Heights (1964)

    MATH  Google Scholar 

  15. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1997)

    Google Scholar 

  16. Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000)

    Article  MathSciNet  Google Scholar 

  17. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximation for solving space fractional diffusion equations. arXiv:1201.5949 [math.NA]

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, F.Z.: Matrix Theory: Basic Results and Techniques, 2nd edn. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

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Acknowledgements

The authors thank Prof Yujiang Wu for his constant encouragement and support. This work was supported by the Program for New Century Excellent Talents in University under Grant No. NCET-09-0438, the National Natural Science Foundation of China under Grant No. 10801067 and No. 11271173, and the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2010-63 and No. lzujbky-2012-k26.

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  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China

    Han Zhou, WenYi Tian & Weihua Deng

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  1. Han Zhou
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  2. WenYi Tian
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  3. Weihua Deng
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Correspondence to Weihua Deng.

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Zhou, H., Tian, W. & Deng, W. Quasi-Compact Finite Difference Schemes for Space Fractional Diffusion Equations. J Sci Comput 56, 45–66 (2013). https://doi.org/10.1007/s10915-012-9661-0

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  • DOI: https://doi.org/10.1007/s10915-012-9661-0

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