Skip to main content
SpringerLink
Log in

Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives

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

Abstract

In this paper, superconvergence points are located for the approximation of the Riesz derivative of order \(\alpha \) using classical Lobatto-type polynomials when \(\alpha \in (0,1)\) and generalized Jacobi functions (GJF) for arbitrary \(\alpha > 0\), respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre–Lobatto expansion. It is observed that the convergence rate for different \(\alpha \) at the superconvergence points is at least \(O(N^{-2})\) better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order \(\alpha > 1\) with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be \(O(N^{-(\alpha +3)/2})\) for \(\alpha \in (0,1)\) and \(O(N^{-2})\) for \(\alpha > 1\). Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Askey, R.: Orthogonal Polynomials and Special Functions. SIAM, Philadelphia (1975)

    MATH  Google Scholar 

  2. Bernstein, S.N.: Sur l’ordre de la meilleure approximation des foncions continues par des polynomes de degré donné. Mém. Publ. Class Sci. Acad. Belgique (2) 4, 1–103 (1912)

    MATH  Google Scholar 

  3. Bu, W., Tang, Y., Yang, J.: Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations. J. Comput. Phys. 276, 26–38 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Chen, F., Xu, Q., Hesthaven, J.S.: A multi-domain spectral method for time-fractional differential equations. J. Comput. Phys. 293, 157–172 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Chen, S., Shen, J., Wang, L.-L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85, 1603–1638 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Davis, P.J.: Interpolation and Approximation. Dover, New York (1975)

    MATH  Google Scholar 

  7. Deng, K., Deng, W.: Finite difference/predictor-corrector approximations for the space and time fractional Fokker–Planck equation. Appl. Math. Lett. 25(11), 1815–1821 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Fatone, L., Funaro, D.: Optimal collocation nodes for fractional derivative operators. SIAM J. Sci. Comput. 37, A1504–A1524 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Huang, C., Zhang, Z., Song, Q.: Spectral methods for substantial fractional differential equations. J. Sci. Comput. 74, 1554–1574 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Ishteva, M., Boyadjiev, L., Scherer, R.: On the Caputo operator of fractional calculus and C-Laguerre functions. Math. Sci. Res. 9, 161–170 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Li, C.P., Zeng, F.H., Liu, F.: Spectral approximations to the fractional integral and derivative. Frac. Calc. Appl. Anal. 15, 383–406 (2012)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. Math. Monogr. Ser. 1. Science Press, Beijing (2006)

    Google Scholar 

  15. Mandelbrot, B.B., Van Ness, J.W.: Fractional brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)

    MathSciNet  MATH  Google Scholar 

  16. Mao, Z., Chen, S., Shen, J.: Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Appl. Numer. Math. 106, 165–181 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Meerschaert, M.M., Benson, D., Baeumer, B.: Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63, 1112–1117 (2001)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Mustapha, K., McLean, W.: Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation. IMA J. Numer. Anal. 32, 906–925 (2012)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \({\mathbb{R}}^2\). J. Comput. Appl. Math. 193(1), 243–268 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science Publishers, Washington (1993)

    MATH  Google Scholar 

  24. Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics, vol. 41. Springer, Berlin (2011)

    MATH  Google Scholar 

  25. Shen, S., Liu, F., Anh, V., Turner, I.: The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. IMA J. Appl. Math. 73, 850–872 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Shen, S., Liu, F., Anh, V., Terner, I., Chen, J.: A novel numerical approximation for the space fractional advection-dispersion equation. IMA J. Appl. Math. 79(3), 421–444 (2014)

    MathSciNet  Google Scholar 

  27. Stynes, M., Gracia, J.L.: A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. IMA J. Numer. Anal. 35, 698–721 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Sun, H.G., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A 388, 4586–4592 (2009)

    Google Scholar 

  29. Wahlbin, L.B.: Superconvergence in Galerkin Finite Element Methods. Lecture Notes in Mathematics, vol. 1605. Springer, Berlin (1995)

    MATH  Google Scholar 

  30. Wang, H., Du, N.: Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J. Comput. Phys. 258, 305–318 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Wang, L.-L., Zhao, X.D., Zhang, Z.: Superconvergence of Jacobi–Gauss-type spectral interpolation. J. Sci. Comput 59, 667–687 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Xie, Z., Wang, L., Zhao, X.: On exponential convergence of Gegenbauer interpolation and spectral differentiation. Math. Comput. 82, 1017–1036 (2012)

    MathSciNet  MATH  Google Scholar 

  33. Xu, Q., Hesthaven, J.S.: Stable multi-domain spectral penalty methods for fractional partial differential equations. J. Comput. Phys. 257, 241–258 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Yang, Q., Turner, I., Liu, F., Ilić, M.: Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions. SIAM J. Sci. Comput. 33, 1159–1180 (2011)

    MathSciNet  MATH  Google Scholar 

  35. Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximations. J. Comput. Phys. 47, 2108–2131 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, A40–A62 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Zeng, F., Liu, F., Li, C.P., Burrage, K., Turner, I., Anh, V.: Crank-Nicolson ADI spectral method for the 2-D Riesz space fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52, 2599–2622 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Zhang, Z.: Superconvergence of a Chebyshev spectral collocation method. J. Sci. Comput. 34, 237–246 (2008)

    MathSciNet  MATH  Google Scholar 

  39. Zhang, Z.: Superconvergence points of polynomial spectral interpolation. SIAM J. Numer. Anal. 50, 2966–2985 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Zhao, X., Sun, Z., Hao, Z.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation. SIAM J. Sci. Comput. 36, 2865–2886 (2014)

    MathSciNet  MATH  Google Scholar 

  41. Zhao, X., Zhang, Z.: Superconvergence points of fractional spectral interpolation. SIAM J. Sci. Comput. 38, A598–A613 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Zheng, M., Liu, F., Turner, I., Anh, V.: A novel high order space-time spectral method for the time-fractional Fokker–Planck equation. SIAM J. Sci. Comput. 37, A701–A724 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Beichuan Deng & Zhimin Zhang

  2. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

  3. School of Mathematics, Southeast University, Nanjing, 210096, China

    Xuan Zhao

Authors
  1. Beichuan Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhimin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhimin Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported in part by the National Natural Science Foundation of China under grants NSFC 11871092, NSAF U1530401, 11701081, 1186010285; the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (No. BM2017002), the Natural Science Youth Foundation of Jiangsu Province of China (Nos. BK20160660); the Fundamental Research Funds for the Central Universities of China (No. 2242019K40111); and Key Project of Natural Science Foundation of China (No. 61833005).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, B., Zhang, Z. & Zhao, X. Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives. J Sci Comput 81, 1577–1601 (2019). https://doi.org/10.1007/s10915-019-01054-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-01054-6

Keywords

Mathematics Subject Classification

Search

Navigation