Skip to main content

Advertisement

SpringerLink
Log in

Numerical correction of finite difference solution for two-dimensional space-fractional diffusion equations with boundary singularity

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

Abstract

In this paper, an efficient algorithm is presented by adopting the extrapolation technique to improve the accuracy of finite difference schemes for two-dimensional space-fractional diffusion equations with non-smooth solution. The popular fractional centered difference scheme is revisited and the stability and error estimation of numerical solution are given in maximum norm. Based on the analysis of leading singularity of exact solution for the underlying problem, the extrapolation technique and numerical correction method are exploited to enhance the accuracy and convergence rate of the computation. Two numerical examples are provided to validate the theoretical prediction and efficiency of the algorithm. It is shown that, by using the proposed algorithm, both accuracy and convergence rate of numerical solutions can be significantly improved and the second-order accuracy can even be recovered for the equations with large fractional orders.

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

Similar content being viewed by others

References

  1. Acosta, G., Borthagaray, J. P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55, 472–495 (2017)

    Article  MathSciNet  Google Scholar 

  2. Antunes, P. R. S., Ferreira, R. A. C.: An augmented-RBF method for solving fractional Sturm-Liouville eigenvalue problems. SIAM J. Sci. Comput. 37, A515–A535 (2015)

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

  4. Chen, X., Zeng, F., Karniadakis, G. E.: A tunable finite difference method for fractional differential equations with non-smooth solutions. Comput. Methods Appl. Mech. Engrg. 318, 193–214 (2017)

    Article  MathSciNet  Google Scholar 

  5. Deng, W., Li, B., Tian, W., Zhang, P.: Boundary problems for the fractional and tempered fractional operators. Multiscale Model Simul. 16, 125–149 (2018)

    Article  MathSciNet  Google Scholar 

  6. Ding, H., Li, C., Chen, Y.: High-order algorithms for Riesz derivative and their applications (II). J. Comput. Phys. 293, 218–237 (2015)

    Article  MathSciNet  Google Scholar 

  7. Epps, B. P., Cushman-Roisin, B.: Turbulence modeling via the fractional Laplacian. arXiv:1803.05286v1 (2018)

  8. Ervin, V. J., Heuer, N., Roop, J. P.: Regularity of the solution to 1-D fractional order diffusion equations. Math. Comp. 87, 2273–2294 (2018)

    Article  MathSciNet  Google Scholar 

  9. Ervin, V. J., Roop, J. P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Meth. Part. Diff. Eq. 22, 558–576 (2006)

    Article  MathSciNet  Google Scholar 

  10. Gatto, P., Hesthaven, J. S.: Numerical approximation of the fractional Laplacian via hp-finite elements, with an application to image denoising. J. Sci. Comput. 65, 249–270 (2015)

    Article  MathSciNet  Google Scholar 

  11. Ghanbari, B., Kumar, S., Kumar, R.: A study of behavior for immune and tumor cells in immunogenetic tumor model with non-singular fractional derivative. Chaos Solitons & Fractals. 133, 109619 (2020)

    Article  MathSciNet  Google Scholar 

  12. Gunzburger, M., Jiang, N., Xu, F.: Analysis and approximation of a fractional Laplacian-based closure model for turbulent flows and its connection to Richardson pair dispersion. Comput. Math. Appl. 75, 1973–2001 (2018)

    Article  MathSciNet  Google Scholar 

  13. Hao, Z., Cao, W.: An improved algorithm based on finite difference schemes for fractional boundary value problems with nonsmooth solution. J. Sci. Comput. 73, 395–415 (2017)

    Article  MathSciNet  Google Scholar 

  14. Hao, Z., Lin, G., Zhang, Z.: Error estimates of a spectral Petrov-Galerkin method for two-sided fractional reaction-diffusion equations. Appl. Math. Comput. 374, 125045 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Hao, Z., Zhang, Z.: Optimal regularity and error estimates of a spectral Galerkin method for fractional advection-diffusion-reaction equations. SIAM J. Numer. Anal. 58, 211–233 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Hao, Z., Zhang, Z., Du, R.: Fractional centered difference scheme for high-dimensional integral fractional Laplacian. Researchgate. https://www.researchgate.net/publication/335888811 (2019)

  18. Jin, B., Zhou, Z.: A finite element method with singularity reconstruction for fractional boundary value problems. ESAIM Math. Model. Numer. Anal. 49, 1261–1283 (2015)

    Article  MathSciNet  Google Scholar 

  19. Kopteva, N., Stynes, M.: An efficient collocation method for a Caputo two-point boundary value problem. BIT. 55, 1105–1123 (2015)

    Article  MathSciNet  Google Scholar 

  20. Kumar, S., Kumar, R., Singh, J., Nisar, K. S., Kumar, D.: An efficient numerical scheme for fractional model of HIV-1 infection of CD4+ T-cells with the effect of antiviral drug therapy. Alexandria Engineering Journal. https://doi.org/10.1016/j.aej.2019.12.046 (2019)

  21. Laskin, N.: Fractional quantum mechanics and lévy path integrals. Phys. Lett. A. 268, 298–305 (2000)

    Article  MathSciNet  Google Scholar 

  22. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods (BAIL 2002) 166, 209–219 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Liu, F., Zhuang, P., Turner, I., Anh, V., Burrage, K.: A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain. J. Comput. Phys. 293, 252–263 (2015)

    Article  MathSciNet  Google Scholar 

  24. Magin, R., Abdullah, O., Baleanu, D., Zhou, X. J.: Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation. J Magn Reson. 190, 255–270 (2008)

    Article  Google Scholar 

  25. 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)

    Article  MathSciNet  Google Scholar 

  26. Song, F., Xu, C.: Spectral direction splitting methods for two-dimensional space fractional diffusion equations. J. Comput. Phys. 299, 196–214 (2015)

    Article  MathSciNet  Google Scholar 

  27. Stynes, M.: Singularities. In: Karniadakis, G. E. (ed.) Handbook of Fractional Calculus with Applications, vol. 3, pp 287–305. De Gruyter, Berlin (2019)

  28. Sun, H., Sun, Z. Z., Gao, G. H.: Some high order difference schemes for the space and time fractional Bloch-Torrey equations. Appl. Math. Comput. 281, 356–380 (2016)

    MathSciNet  MATH  Google Scholar 

  29. 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  Google Scholar 

  30. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Woyczyński, W. A.: Lévy Processes in the Physical Sciences. In: Barndorff-Nielsen, O.E., Resnick, S. I., Mikosch, T. (eds.) Processes, Lévy, pp 241–266. Birkhäuser, Boston (2001)

  33. Xu, C.: Spectral methods for some kinds of fractional differential equations: traditional and Müntz spectral methods. In: Karniadakis, G. E. (ed.) Handbook of Fractional Calculus with Applications, vol. 3, pp 101–126. De Gruyter, Berlin (2019)

  34. Zayernouri, M., Karniadakis, G. E.: Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)

    Article  MathSciNet  Google Scholar 

  35. Zhao, L., Deng, W.: High order finite difference methods on non-uniform meshes for space fractional operators. Adv. Comput. Math. 42, 425–468 (2016)

    Article  MathSciNet  Google Scholar 

  36. Zhao, X., Sun, Z. Z., Hao, Z.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation. SIAM J. Sci. Comput. 36, A2865–A2886 (2014)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Prof. Zhi-Zhong Sun and Prof. Rui Du for helpful discussion and suggestion.

Funding

This work was partially supported by the NSF of China (No. 11671083) and the Fundamental Research Funds for the Central Universities.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, 01609, USA

    Zhaopeng Hao

  2. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Wanrong Cao & Shengyue Li

Authors
  1. Zhaopeng Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wanrong Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shengyue Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wanrong Cao.

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

Hao, Z., Cao, W. & Li, S. Numerical correction of finite difference solution for two-dimensional space-fractional diffusion equations with boundary singularity. Numer Algor 86, 1071–1087 (2021). https://doi.org/10.1007/s11075-020-00923-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00923-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation