Skip to main content
SpringerLink
Log in

Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations

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

Abstract

Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian elimination requires \(\mathcal {O}\left ({NM}_{1}M_{2}\left ({M_{1}^{2}}+{M_{2}^{2}}+NM_{1}M_{2}\right )\right )\) complexity and \(\mathcal {O}\left ({N{M_{1}^{2}}{M_{2}^{2}}}\right )\) storage, where N is the number of temporal unknown and M1, M2 are the numbers of spatial unknown in x, y directions respectively. By exploring the structure of the coefficient matrix in fully coupled form, it possesses block lower-triangular Toeplitz structure and its blocks are block-dense Toeplitz matrices with dense-Toeplitz blocks. Based on this special structure and cooperating with time-marching or divide-and-conquer technique, two fast solvers with storage \(\mathcal {O}\left ({NM}_{1}M_{2}\right )\) are developed. The complexity for the fast solver via time-marching is \(\mathcal {O}\left ({NM}_{1}M_{2}\left (N+\log \left (M_{1}M_{2}\right )\right )\right )\) and the one via divide-and-conquer technique is \(\mathcal {O}\left ({NM}_{1}M_{2}\left (\log ^{2} N+ \log \left (M_{1}M_{2}\right )\right )\right )\). It is worth to remark that the proposed solvers are not lossy. Some discussions on achieving convergence rate for smooth and non-smooth solutions are given. Numerical results show the high efficiency of the proposed fast solvers.

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

Similar content being viewed by others

References

  1. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertaccini, D., Durastante, F.: Limited memory block preconditioners for fast solution of fractional partial differential equations. J. Sci. Comput. 77(2), 950–970 (2018). https://doi.org/10.1007/s10915-018-0729-3

    Article  MathSciNet  MATH  Google Scholar 

  3. Breiten, T., Simoncini, V., Stoll, M.: Low-rank solvers for fractional differential equations. Electron. T. Numer. Anal. 45, 107–132 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Capizzano, S.S., Tyrtyshnikov, E.: Any circulant-like preconditioner for multilevel matrices is not superlinear. SIAM J. Matrix Anal. Appl. 21(2), 431–439 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chan, R.H., Ng, M.K.: Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38(3), 427–482 (1996). https://doi.org/10.1137/s0036144594276474

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, X., Wang, W.F., Ding, D., Lei, S.L.: A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation. Comput. Math. Appl. 73(9), 1932–1944 (2017). https://doi.org/10.1016/j.camwa.2017.02.040

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Deng, W.H.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47(1), 204–226 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys 307, 262–279 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ford, N.J., Yan, Y.: An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data. Fract. Calc. Appl. Anal. 20(5), 1076–1105 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fu, H.F., Ng, M.K., Wang, H.: A divide-and-conquer fast finite difference method for space-time fractional partial differential equation. Comput. Math. Appl. 73(6), 1233–1242 (2017). https://doi.org/10.1016/j.camwa.2016.11.023

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, G.H., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230(3), 586–595 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gohberg, I., Olshevsky, V.: Circulants, displacements and decompositions of matrices. Integr. Equat. Oper. Th. 15(5), 730–743 (1992). https://doi.org/10.1007/bf01200697

    Article  MathSciNet  MATH  Google Scholar 

  14. Hao, Z., Cao, W.: An improved algorithm based on finite difference schemes for fractional boundary value problems with nonsmooth solution. J. Sci. Comput. 73(1), 395–415 (2017). https://doi.org/10.1007/s10915-017-0417-8

    Article  MathSciNet  MATH  Google Scholar 

  15. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  16. Huang, Y.C., Lei, S.L.: A fast numerical method for block lower triangular Toeplitz with dense Toeplitz blocks system with applications to time-space fractional diffusion equations. Numer. Algor. 76(3), 605–616 (2017). https://doi.org/10.1007/s11075-017-0272-6

    Article  MathSciNet  MATH  Google Scholar 

  17. Jiang, S.D., Zhang, J.W., Zhang, Q., Zhang, Z.M.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21(3), 650–678 (2017). https://doi.org/10.4208/cicp.OA-2016-0136

    Article  MathSciNet  Google Scholar 

  18. Jin, B., Lazarov, R., Zhou, Z.: Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview. Comput. Methods Appl. Mech. Eng. 346, 332–358 (2019)

    Article  MathSciNet  Google Scholar 

  19. Ke, R.H., Ng, M.K., Sun, H.W.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations. J. Comput. Phys. 303, 203–211 (2015). https://doi.org/10.1016/j.jcp.2015.09.042

    Article  MathSciNet  MATH  Google Scholar 

  20. Kopteva, N., Stynes, M.: Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem. Adv. Comput. Math. 43(1), 77–99 (2017). https://doi.org/10.1007/s10444-016-9476-x

    Article  MathSciNet  MATH  Google Scholar 

  21. Langlands, T., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205(2), 719–736 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lei, S.L., Huang, Y.C.: Fast algorithms for high-order numerical methods for space fractional diffusion equations. Int. J. Comput. Math. 94(5), 1062–1078 (2017). https://doi.org/10.1080/00207160.2016.1149579

    Article  MathSciNet  MATH  Google Scholar 

  23. Lei, S.L., Sun, H.W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013). https://doi.org/10.1016/j.jcp.2013.02.025

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, F.W., Zhuang, P.H., Turner, I., Burrage, K., Anh, V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Modell. 38(15), 3871–3878 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lu, X., Pang, H.K., Sun, H.W.: Fast approximate inversion of a block triangular Toeplitz matrix with applications to fractional sub-diffusion equations. Numer. Linear Algebra Appl. 22(5), 866–882 (2015). https://doi.org/10.1002/nla.1972

    Article  MathSciNet  MATH  Google Scholar 

  26. Lu, X., Pang, H.K., Sun, H.W., Vong, S.: Approximate inversion method for time-fractional subdiffusion equations. Numer. Linear Algebra Appl. https://doi.org/10.1002/nla.2132

  27. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Redding, Danbury (2006)

    Google Scholar 

  29. Moghaderi, H., Dehghan, M., Donatelli, M., Mazza, M.: Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations. J. Comput. Phys 350, 992–1011 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pan, J.Y., Ke, R.H., Ng, M.K., Sun, H.W.: Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations. SIAM J. Sci. Comput. 36(6), A2698–A2719 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pang, H.K., Sun, H.W.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231(2), 693–703 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pang, H.K., Sun, H.W.: Fourth order finite difference schemes for time–space fractional sub-diffusion equations. Comput. Math. Appl. 71(6), 1287–1302 (2016). https://doi.org/10.1016/j.camwa.2016.02.011

    Article  MathSciNet  Google Scholar 

  33. Podlubny, I.: Fractional Differential Equations, vol. 198. Academic Press, New York (1999)

    MATH  Google Scholar 

  34. Song, J., Yu, Q., Liu, F.W., Turner, I.: A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation. Numer. Algor. 66(4), 911–932 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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). https://doi.org/10.1016/j.amc.2016.01.044

    Article  MathSciNet  MATH  Google Scholar 

  36. Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yan, Y., Sun, Z.Z., Zhang, J.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22(4), 1028–1048 (2017). https://doi.org/10.4208/cicp.OA-2017-0019

    Article  MathSciNet  Google Scholar 

  38. Yu, Q., Liu, F.W., Turner, I., Burrage, K.: A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D. Appl. Math. Comput. 219(8), 4082–4095 (2012). https://doi.org/10.1016/j.amc.2012.10.056

    Article  MathSciNet  MATH  Google Scholar 

  39. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40–A62 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zeng, F., Mao, Z., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. SIAM J. Sci. Comput. 39(1), A360–A383 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zeng, F., Turner, I., Burrage, K.: A stable fast time-stepping method for fractional integral and derivative operators. J. Sci. Comput. https://doi.org/10.1007/s10915-018-0707-9 (2018)

  42. Zeng, F., Zhang, Z., Karniadakis, G.E.: Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Comput. Methods Appl. Mech. Eng. 327, 478–502 (2017)

    Article  MathSciNet  Google Scholar 

  43. Zhao, L., Deng, W.: High order finite difference methods on non-uniform meshes for space fractional operators. Adv. Comput. Math. 42(2), 425–468 (2016). https://doi.org/10.1007/s10444-015-9430-3

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Macau, Macau, China

    Yun-Chi Huang & Siu-Long Lei

Authors
  1. Yun-Chi Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siu-Long Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siu-Long Lei.

Additional information

Publisher’s note

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

This research was partially supported by the research grants MYRG2016-00202-FST, MYRG2018-00025-FST from University of Macau and 048/2017/A from Macao Science and Technology Development Fund (FDCT).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, YC., Lei, SL. Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations. Numer Algor 84, 37–62 (2020). https://doi.org/10.1007/s11075-019-00742-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00742-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation