PCG method with Strang’s circulant preconditioner for Hermitian positive definite linear system in Riesz space fractional advection–dispersion equations



In this paper, preconditioned conjugate gradient (PCG) method with Strang’s circulant preconditioner is investigated to solve the Hermitian positive definite linear systems, which is result from the Crank–Nicolson (C-N) finite difference scheme with the weighted and shifted Grünwald difference (WSGD) operators to discretize the Riesz space fractional advection–dispersion equation (RSFADE). We show that the spectrum of the preconditioned matrix is clustered around 1, and the singular values of the preconditioned matrix are uniformly bounded away from zero under a certain condition, respectively; hence the PCG method, when applied to solving the preconditioned system, converges superlinearly. Moreover, the complexity in each iteration of the PCG method is \(O(N\log N)\) via using the fast Fourier transforms, where N is the matrix size. Numerical experiments are included to demonstrate the effectiveness of our approach.


Preconditioned conjugate gradient method Riesz space fractional advection–dispersion equation Crank–Nicolson finite difference scheme Strang’s circulant preconditioner Fast Fourier transforms 

Mathematics Subject Classification

26A33 34A08 65L12 65F08 65F10 65T50 



The authors would like to convey their thanks to the anonymous referees and the associate editor for their very helpful comments and suggestions, which greatly improved this paper.


  1. Bai J, Feng X (2007) Fractional-order anisotropic diffusion for image denoising. IEEE Trans Image Proc 16:2492–2502MathSciNetCrossRefGoogle Scholar
  2. Benson D, Wheatcraft SW, Meerschaert MM (2000a) Application of a fractional advection–dispersion equation. Water Resour Res 36:1403–1413CrossRefGoogle Scholar
  3. Benson D, Wheatcraft SW, Meerschaert MM (2000b) The fractional-order governing equation of Lévy motion. Water Resour. Res 36:1413–1423CrossRefGoogle Scholar
  4. Carella AR, Dorao CA (2013) Least-squares spectral method for the solution of a fractional advection–dispersion equation. J Comput Phys 232:33–45MathSciNetCrossRefGoogle Scholar
  5. Carreras BA, Lynch VE, Zaslavsky GM (2001) Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models. Phys Plasma 8:5096–5103CrossRefGoogle Scholar
  6. Çelik C, Duman M (2012) Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J Comput Phys 231:1743–1750MathSciNetCrossRefMATHGoogle Scholar
  7. Chan R, Jin XQ (2007) An introduction to iterative Toeplitz solvers. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  8. Chan R, Ng MK (1996) Conjugate gradient methods for Toeplitz systems. SIAM Rev 38:427–482MathSciNetCrossRefMATHGoogle Scholar
  9. Chan R, Strang G (1989) Toeplitz equations by conjugate gradients with circulant preconditioner. SIAM J Sci Stat Comput 10:104–119MathSciNetCrossRefMATHGoogle Scholar
  10. Chen W, Holm S (2004) Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency dependency. J Acoust Soc Am 115(4):1424–1430CrossRefGoogle Scholar
  11. Chen W, Pang GF (2016) A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction. J Comput Phys 309(15):350–367MathSciNetCrossRefMATHGoogle Scholar
  12. Chen W, Hu S, Cai W (2016) A causal fractional derivative model for acoustic wave propagation in lossy media. Arch Appl Mech 86(3):529–539CrossRefGoogle Scholar
  13. Chen W, Fang J, Pang GF, Holm S (2017) Fractional biharmonic operator equation model for arbitrary frequency-dependent scattering attenuation in acoustic wave propagation. J Acoust Soc Am 141(1):244–253CrossRefGoogle Scholar
  14. Chou LK, Lei SL (2016) Fast ADI method for high dimensional fractional diffusion equations in conservative form with preconditioned strategy. Comput Math Appl 73(3):385–403MathSciNetCrossRefMATHGoogle Scholar
  15. Ding HF, Li CP (2016) High-order algorithms for Riesz derivative and their applications (III). Fract Calc Appl Anal 19:19–55MathSciNetCrossRefMATHGoogle Scholar
  16. Ding HF, Li CP, Chen YQ (2014) High-order algorithms for Riesz derivative and their applications (I). Abstr Appl Anal 17.
  17. Ding HF, Li CP, Chen YQ (2015) High-order algorithms for Riesz derivative and their applications (II). J Comput Phys 293:218–237MathSciNetCrossRefMATHGoogle Scholar
  18. Ervin VJ, Roop JP (2006) Variational formulation for the stationary fractional advection–dispersion equation. Numer Methods Partial Differ Equ 22:558–576MathSciNetCrossRefMATHGoogle Scholar
  19. Feng LB, Zhuang PH, Liu FW, Turner I, Li J (2016) High-order numerical methods for the Riesz space fractional advection–dispersion equations. Comput Math Appl.
  20. Golbabai A, Sayevand K (2011) Analytical modelling of fractional advection–dispersion equation defined in a bounded space domain. Math Comput Model 53:1708–1718MathSciNetCrossRefMATHGoogle Scholar
  21. Hejazi H, Moroney T, Liu FW (2014) Stability and convergence of a finite volume method for the space fractional advection–dispersion equation. J Comput Appl Math 255:684–697MathSciNetCrossRefMATHGoogle Scholar
  22. Horn RA, Johnson CR (1991) Topic in matrix analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  23. Huang FH, Liu FW (2005) The fundamental solution of the space-time fractional advection–dispersion equation. J Appl Math Comput 18(1–2):339–350MathSciNetCrossRefMATHGoogle Scholar
  24. Jin XQ, Lin FR, Zhao Z (2015) Preconditioned iterative methods for two-dimensional space-fractional diffusion equations. Commun Comput Phys 18(2):469–488MathSciNetCrossRefMATHGoogle Scholar
  25. Lei SL, Sun HW (2013) A circulant preconditioner for fractional diffusion equations. J Comput Phys 242:715–725MathSciNetCrossRefMATHGoogle Scholar
  26. Lei SL, Chen X, Zhang XH (2016) Multilevel circulant preconditioner for high-dimensional fractional diffusion equations. East Asian J Appl Math 6(2):109–130MathSciNetCrossRefMATHGoogle Scholar
  27. Lin FR, Yang SW, Jin XQ (2014) Preconditioned iterative methods for fractional diffusion equation. J Comput Phys 256:109–117MathSciNetCrossRefMATHGoogle Scholar
  28. Liu FW, Anh V, Turner I, Zhuang PH (2003) Time fractional advection–dispersion equation. J Appl Math Comput 13(1–2):233–246MathSciNetCrossRefMATHGoogle Scholar
  29. Liu FW, Zhuang PH, Anh V, Turner I, Burra K (2007a) Stability and convergence of the difference methods for the space-time fractional advection–diffusion equation. Appl Math Comput 191:12–20MathSciNetMATHGoogle Scholar
  30. Liu QX, Liu FW, Turner I, Anh V (2007b) Approximation of the Levy–Feller advection–dispersion process by random walk and finite difference method. J Comput Phys 222(1):57–70MathSciNetCrossRefMATHGoogle Scholar
  31. Magin RL (2006) Fractional calculus in bioengineering. Begell House Publishers, New YorkGoogle Scholar
  32. Meerschaert MM, Tadjeran C (2004) Finite difference approximations for fractional advection–dispersion flow equations. J Comput Appl Math 172:65–77MathSciNetCrossRefMATHGoogle Scholar
  33. Meerschaert MM, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 56:80–90MathSciNetCrossRefMATHGoogle Scholar
  34. Metzler R, Klafter J (2000) The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77MathSciNetCrossRefMATHGoogle Scholar
  35. Pan JY, Ke RH, Ng MK, Sun HW (2014) Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations. SIAM J Sci Comput 36:A2698–A2719MathSciNetCrossRefMATHGoogle Scholar
  36. Pang HK, Sun HW (2012) Multigrid method for fractional diffusion equations. J Comput Phys 231:693–703MathSciNetCrossRefMATHGoogle Scholar
  37. Pang HK, Sun HW (2016) Fast numerical contour integral method for fractional diffusion equations. J Sci Comput 66:41–66MathSciNetCrossRefMATHGoogle Scholar
  38. Qu W, Lei SL, Vong SW (2014) Circulant and skew-circulant splitting iteration for fractional advection–diffusion equations. Int J Comput Math 91:2232–2242MathSciNetCrossRefMATHGoogle Scholar
  39. Raberto M, Scalas E, Mainardi F (2002) Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314:749–755CrossRefMATHGoogle Scholar
  40. Ran YH, Wang JG (2016) On ADI-like iteration method for fractional diffusion equations. Linear Algebra Appl 493:544–555MathSciNetCrossRefMATHGoogle Scholar
  41. Saichev AI, Zaslavsky GM (1997) Fractional kinetic equations: solutions and applications. Chaos 7:753–764MathSciNetCrossRefMATHGoogle Scholar
  42. Shen SJ, Liu FW, Anh V, Turner I (2008) The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation. IMA J Appl Math 73(6):850–872MathSciNetCrossRefMATHGoogle Scholar
  43. Shen SJ, Liu FW, Anh V (2011) Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection–diffusion equation. Numer Algorithms 56:383–403MathSciNetCrossRefMATHGoogle Scholar
  44. Shlesinger MF, West BJ, Klafter J (1987) Lévy dynamics of enhanced diffusion: application to turbulence. Phys Rev Lett 58:1100–1103MathSciNetCrossRefGoogle Scholar
  45. Sousa E (2009) Finite difference approximates for a fractional advection diffusion problem. J Comput Phys 228:4038–4054MathSciNetCrossRefMATHGoogle Scholar
  46. Sousa E, Li C (2015) A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville derivative. Appl Numer Math 90:22–37MathSciNetCrossRefMATHGoogle Scholar
  47. Tian WY, Zhou H, Deng WH (2015) A class of second order difference approximation for solving space fractional diffusion equations. Math Comput 84:1703–1727MathSciNetCrossRefMATHGoogle Scholar
  48. Wang H, Du N (2013) A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation. J Comput Phys 253:50–63MathSciNetCrossRefMATHGoogle Scholar
  49. Wang H, Du N (2014) Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J Comput Phys 258:305–318MathSciNetCrossRefMATHGoogle Scholar
  50. Wang KQ, Wang H (2011) A fast characteristic finite difference method for fractional advection–diffusion equations. Adv Water Resour 34:810–816CrossRefGoogle Scholar
  51. Wang WF, Chen X, Ding D, Lei SL (2015a) Circulant preconditioning technique for barrier options pricing under fractional diffusion models. Int J Comput Math 92(12):2596–2614MathSciNetCrossRefMATHGoogle Scholar
  52. Wang XB, Liu FW, Chen XJ (2015b) Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations. Adv Math Phys. 14.
  53. Zaslavsky GM (2002) Chaos, fractional kinetics, and anomalous transport. Phys Rep 371:461–580MathSciNetCrossRefMATHGoogle Scholar
  54. Zhang L, Sun HW, Pang HK (2015) Fast numerical solution for fractional diffusion equations by exponential quadrature rule. J Comput Phys 299:130–143MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShaoguan UniversityShaoguanChina
  2. 2.State Key Laboratory of Quality Research in Chinese Medicines and Faculty of Information TechnologyMacau University of Science and TechnologyTaipaChina

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