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A τ-preconditioner for a non-symmetric linear system arising from multi-dimensional Riemann-Liouville fractional diffusion equation

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Abstract

In this paper, we study a τ-preconditioner for non-symmetric linear system arising from a steady-state multi-dimensional Riemann-Liouville (RL) fractional diffusion equation. The generalized minimal residual (GMRES) method is applied to solve the preconditioned linear system. Theoretically, we show that the GMRES solver for the preconditioned linear system has a convergence rate independent of discretization stepsizes. To the best of our knowledge, this is the first iterative solver with stepsize-independent convergence rate for the non-symmetric linear system. The proposed τ-preconditioner is diagonalizable by the sine transform matrix, thanks to which the matrix-vector multiplication in each iteration step can be fast implemented by the fast sine transform (FST). Hence, the total operation cost of the proposed solver for the non-symmetric problem is linearithmic. Numerical results are reported to show the efficiency of the proposed preconditioner.

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The data of codes involved in this paper are available from the corresponding author on reasonable request.

References

  1. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36(6), 1403–1412 (2000)

    Article  Google Scholar 

  2. Bini, D., Benedetto, F.: A new preconditioner for the parallel solution of positive definite toeplitz systems. In: Proceedings of the second annual ACM Symposium on Parallel Algorithms and Architectures, pp. 220–223 (1990)

  3. del Castillo-Negrete, D., Carreras, B., Lynch, V.: Fractional diffusion in plasma turbulence. Physics of Plasmas 11(8), 3854–3864 (2004)

    Article  Google Scholar 

  4. Çelik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, M.H., Deng, W.H.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, W.: A speculative study of 2/ 3-order fractional laplacian modeling of turbulence: some thoughts and conjectures. Chaos:, An Interdisciplinary Journal of Nonlinear Science 16(2), 023, 126 (2006)

    Article  Google Scholar 

  7. Dai, P., Wu, Q., Wang, H., Zheng, X.: An efficient matrix splitting preconditioning technique for two-dimensional unsteady space-fractional diffusion equations. J. Comput. Appl. Math. 371, 112, 673 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics Numerical Mathematics and Scie (2014)

  10. Gu, X.M., Huang, T.Z., Zhao, Y.L., Lyu, P., Carpentieri, B.: A fast implicit difference scheme for solving the generalized time–space fractional diffusion equations with variable coefficients. Numer. Methods Partial Differ. Equ. 37(2), 1136–1162 (2021)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, X., Lin, X.L., Ng, M.K., Sun, H.W.: Spectral analysis for preconditioning of multi-dimensional riesz fractional diffusion equations. arXiv:2102.01371 (2021)

  13. Jia, J., Wang, H., Zheng, X.: A fast collocation approximation to a two-sided variable-order space-fractional diffusion equation and its analysis. J. Comput. Appl. Math. 388, 113, 234 (2021)

    Article  MathSciNet  Google Scholar 

  14. Jia, J., Zheng, X., Fu, H., Dai, P., Wang, H.: A fast method for variable-order space-fractional diffusion equations. Numer. Algorithms 85(4), 1519–1540 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jian, H.Y., Huang, T.Z., Zhao, X.L., Zhao, Y.L.: A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients. Adv. Differ. Equ 2018(1), 1–24 (2018)

    Article  MathSciNet  Google Scholar 

  16. Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comput. 84(296), 2665–2700 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jin, X.Q.: Preconditioning techniques for Toeplitz systems Higher Education Press (2010)

  18. Jin, X.Q., Lin, F.R., Zhao, Z.: Preconditioned iterative methods for two-dimensional space-fractional diffusion equations. Commun. Comput. Phys. 18, 469–488 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Lin, X.l., Ng, M.K., Sun, H.W.: A splitting preconditioner for toeplitz-like linear systems arising from fractional diffusion equations. SIAM J. Matrix Anal. Appl. 38(4), 1580–1614 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, Q., Liu, F.W., Gu, Y.T., Zhuang, P.H., Chen, J., Turner, I.: A meshless method based on point interpolation method (PIM) for the space fractional diffusion equation. Appl. Math. Comput. 256, 930–938 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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 

  25. Mittnik, S., Rachev, S.: Option pricing for stable and infinitely divisible asset returns. Math. Comput. Model. 29(10-12), 93–104 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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, A2698–A2719 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Pang, H.K., Sun, H.W.: Fast numerical contour integral method for fractional diffusion equations. J. Sci. Comput. 66(1), 41–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives: theory and applications switzerland: gordon and breach science publishers (1993)

  30. Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville derivative. Appl. Numer. Math. 90, 22–37 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comp. 84, 1703–1727 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This research is supported by RGC GRF 12300519, 17201020, and 17300021.

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Authors and Affiliations

  1. Shenzhen JL Computational Science and Applied Research Institute, Shenzhen, People’s Republic of China

    Xue-lei Lin

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

    Xue-lei Lin

  3. Department of Mathematics, University of Macau, Macau, Macao

    Xin Huang & Hai-Wei Sun

  4. Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

    Michael K. Ng

Authors
  1. Xue-lei Lin
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  2. Xin Huang
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  3. Michael K. Ng
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  4. Hai-Wei Sun
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Correspondence to Michael K. Ng.

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Lin, Xl., Huang, X., Ng, M.K. et al. A τ-preconditioner for a non-symmetric linear system arising from multi-dimensional Riemann-Liouville fractional diffusion equation. Numer Algor 92, 795–813 (2023). https://doi.org/10.1007/s11075-022-01342-7

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  • DOI: https://doi.org/10.1007/s11075-022-01342-7

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