Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T12:01:45.697Z Has data issue: false hasContentIssue false
  • English
  • Français

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Part of: Real functions Numerical linear algebra Numerical methods in Fourier analysis Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  12 May 2016

Siu-Long Lei ,
Xu Chen  and
Xinhe Zhang
Siu-Long Lei*
Affiliation:
Department of Mathematics, University of Macau, Macau, China
Xu Chen*
Affiliation:
Department of Mathematics, University of Macau, Macau, China
Xinhe Zhang*
Affiliation:
Department of Mathematics, University of Macau, Macau, China
*
*Corresponding author. Email addresses:sllei@umac.mo (S.-L. Lei), 09xchen@gmail.com (X. Chen), yb57010@connect.umac.mo (X. Zhang)
*Corresponding author. Email addresses:sllei@umac.mo (S.-L. Lei), 09xchen@gmail.com (X. Chen), yb57010@connect.umac.mo (X. Zhang)
*Corresponding author. Email addresses:sllei@umac.mo (S.-L. Lei), 09xchen@gmail.com (X. Chen), yb57010@connect.umac.mo (X. Zhang)
Get access

Abstract

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

Keywords

High-dimensional two-sided fractional diffusion equationimplicit finite difference methodunconditionally stablemultilevel circulant preconditionerGMRES method

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65L12: Finite difference methods 65T50: Discrete and fast Fourier transforms 26A33: Fractional derivatives and integrals
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 2 , May 2016 , pp. 109 - 130
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bai, J. and Feng, X., Fractional-order anisotropic diffusion for image denoising, IEEE Tran. Image Proc. 16, 2492–2502 (2007).CrossRefGoogle ScholarPubMed
[2]Benson, D., Wheatcraft, S.W. and Meerschaert, M.M., Application of a fractional advection-dispersion equation, Water Resour. Res. 36, 1403–1413 (2000).CrossRefGoogle Scholar
[3]Benson, D., Wheatcraft, S.W. and Meerschaert, M.M., The fractional-order governing equation of Lévy motion, Water Resour. Res. 36, 1413–1423 (2000).CrossRefGoogle Scholar
[4]Benzi, M., Preconditioning techniques for large linear systems: A survey, J. Comp. Phys. 182, 418–477 (2002).CrossRefGoogle Scholar
[5]Carreras, B.A., Lynch, V.E. and Zaslavsky, G.M., Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models, Phys. Plasma 8, 5096–5103 (2001).CrossRefGoogle Scholar
[6]Chan, R. and Jin, X., A family of block preconditioners for block systems, SIAM J. Sci. Stat. Comp. 13, 1218–1235 (1992).CrossRefGoogle Scholar
[7]Chan, R. and Jin, X., An introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia (2007).CrossRefGoogle Scholar
[8]Chan, R. and Ng, M., Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38, 427–482 (1996).CrossRefGoogle Scholar
[9]Chan, R. and Strang, G., Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Statist. Comp. 10, 104–119 (1989).CrossRefGoogle Scholar
[10]Chan, T., An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comp. 9, 766–771 (1988).CrossRefGoogle Scholar
[11]Chen, M., Deng, W. and Wu, Y., Superlinearly convergent algorithms for the two-dimensional space-time Caputo-Riesz fractional diffusion equation, Appl. Num. Math. 70, 22–41 (2013).CrossRefGoogle Scholar
[12]Chen, M. and Deng, W., Fourth-order accurate scheme for the space fractional diffusion equations, SIAM J. Num. Anal. 52, 1418–1438 (2014).CrossRefGoogle Scholar
[13]Chen, S., Liu, F., Jiang, X., Turner, I. and Anh, V., A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients, Appl. Math. Comp. 257, 591–601 (2015).CrossRefGoogle Scholar
[14]Davis, P., Circulant Matrices, John Wiley & Sons, New York (1979).Google Scholar
[15]Deng, W., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Num. Anal. 47, 204–226 (2008).CrossRefGoogle Scholar
[16]Du, N. and H. Wang, , A fast finite element method for space-fractional dispersion equations on bounded domains in ℝ2, SIAM J. Sci. Comp. 37), A1614-A1635 (2015).CrossRefGoogle Scholar
[17]Ervin, V.J., Heuer, N.and Roop, J.P., Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Num. Anal. 45, 572–591 (2007).CrossRefGoogle Scholar
[18]Jin, X., Preconditioning Techniques for Teoplitz Systems, Higher Education Press, Beijing (2010).Google Scholar
[19]Jin, X., A note on preconditioned block Toeplitz matrices, SIAM J. Sci. Comp. 16, 951–955 (1995).CrossRefGoogle Scholar
[20]Langlands, T.A.M. and Henry, B.I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Phys. 205, 719–736 (2005).CrossRefGoogle Scholar
[21]Langland, T.A.M., Henry, B.I. and Wearne, S.L., Fractional cable equation models for anomalous electrodiffusion equation in nerve cells: Infinite domain solutions, J. Math. Biol. 59, 761–808 (2009).CrossRefGoogle Scholar
[22]Lei, S. and Sun, H., A circulant preconditioner for fractional diffusion equations, J. Comp. Phys. 242), 715–725 (2013).CrossRefGoogle Scholar
[23]Liu, F., Anh, V. and Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 166, 209–219 (2004).CrossRefGoogle Scholar
[24]Liu, F., Turner, I., Anh, V., Yang, Q. and Burrage, K., A numerical methodfor the fractional Fitzhugh-Nagumo monodomain model, ANZIAM J. 54, C608-C629 (2013).CrossRefGoogle Scholar
[25]Lin, F., Yang, S. and Jin, X., Preconditioned iterative methods for fractional diffusion equation, J. Comp. Phys. 256, 109–117 (2014).CrossRefGoogle Scholar
[26]Li, X. and Xu, C., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comp. Phys. 8, 1016–1051 (2010).Google Scholar
[27]Magin, R.L., Fractional Calculus in Bioengineering, Begell House Publishers, (2006).Google Scholar
[28]Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, Carpinteri, A. and Mainardi, F. (Eds.), pp. 291–348, Springer, Wien (1997).CrossRefGoogle Scholar
[29]Meerschaert, M.M., Scheffler, H.P. and Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comp. Phys. 211, 249–261 (2006).CrossRefGoogle Scholar
[30]Meerschaert, M.M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comp. Appl. Math. 172, 65–77 (2004).CrossRefGoogle Scholar
[31]Meerschaert, M.M. and Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Num. Math. 56, 80–90 (2006).CrossRefGoogle Scholar
[32]Metzler, R. and Klafter, J., The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1–77 (2000).CrossRefGoogle Scholar
[33]Murio, D.A., Implicit finite difference approximation for time fractional diffusion equations, Comp. Math. Appl. 56, 1138–1145 (2008).CrossRefGoogle Scholar
[34]Ng, M., Iterative Methods for Toeplitz Systems, Oxford (2004).CrossRefGoogle Scholar
[35]Oldham, K.B., Fractional differential equations in electrochemistry, Adv. Eng. Software 41, 9–12 (2010).CrossRefGoogle Scholar
[36]Pang, H. and Sun, H., Multigrid method for fractional diffusion equations, J. Comp. Phys. 231, 693–703 (2012).CrossRefGoogle Scholar
[37]Podlubny, I., Fractional Differential Equations, Academic Press, New York (1999).Google Scholar
[38]Raberto, M., Scalas, E. and Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314, 749–755 (2002).CrossRefGoogle Scholar
[39]Ramos-Fernandez, G., Mateos, J., Miramontes, O., Cocho, G., Larralde, H. and Ayala-Orozco, B., Levy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi), Behav. Ecol. Sociobiol. 55, 223–230 (2004).Google Scholar
[40]Ritchie, K., Shan, X.-Y., Kondo, J., Iwasawa, K., Fujiwara, T. and Kusumi, A.Detection of non-Brownian diffusion in the cell membrane in single molecule tracking, Biophysical J. 88, 2266–2277 (2005).CrossRefGoogle ScholarPubMed
[41]Roop, J.P., Computational aspects ofFEM approximation of fractional advection dispersion equations on bounded domains in ℝ2, J. Comp. Appl. Math. 193, 243–268 (2006).CrossRefGoogle Scholar
[42]Qu, W., Lei, S. and Vong, S., Circulant and skew-circulant splitting iteration for fractional advection-diffusion equations, Inter. J. Comp. Math. 91, 2232–2242 (2014).CrossRefGoogle Scholar
[43]Saad, Y., Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, (2003).CrossRefGoogle Scholar
[44]Scher, H. and Montroll, E.W., Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12, 2455–2477 (1975).CrossRefGoogle Scholar
[45]Song, F. and Xu, C., Spectral direction splitting methods for two-dimensional space fractional diffusion equations, J. Comp. Phys. 299, 196–214 (2015).CrossRefGoogle Scholar
[46]Shlesinger, M.F., West, B.J. and Klafter, J., Lévy dynamics of enhanced diffusion: application to turbulence, Phys. Rev. Letts. 58, 1100–1103 (1987).CrossRefGoogle ScholarPubMed
[47]Sokolov, I.M., Klafter, J. and Blumen, A., Fractional kinetics, Phys. Today, 28–53 (Nov., 2002).Google Scholar
[48]Sousa, E. and Li, C., A weighted finite difference methodfor the fractional diffusion equation based on the Riemann-Liouville derivative, Appl. Num. Math. 90, 22–37 (2015).CrossRefGoogle Scholar
[49]Sousa, E., Finite difference approximates for a fractional advection diffusion problem, J. Comp. Phys. 228, 4038–4054 (2009).CrossRefGoogle Scholar
[50]Su, L., Wang, W. and Yang, Z., Finite difference approximations for the fractional advection-diffusion equation, Phys. Letts. A 373, 4405–4408 (2009).CrossRefGoogle Scholar
[51]Tadjeran, C., Meerschaert, M.M. and Scheffler, H.P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comp. Phys. 213, 205–213 (2006).CrossRefGoogle Scholar
[52]Tian, W., Zhou, H. and Deng, W., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp. 84, 1703–1727 (2015).CrossRefGoogle Scholar
[53]Hao, Z., Sun, Z. and Cao, W., A fourth-order approximation of fractional derivatives with its applications, J. Comp. Phys. 281, 787–805 (2015).CrossRefGoogle Scholar
[54]Wang, H. and Du, N., A fast finite difference methodfor three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation, J. Comp. Phys. 253, 50–63 (2013).CrossRefGoogle Scholar
[55]Wang, H. and Du, N., Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations, J. Comp. Phys. 258, 305–318 (2014).CrossRefGoogle Scholar
[56]Wang, H., Wang, K. and Sircar, T., A direct O(N log2N) finite difference method for fractional diffusion equations, J. Comp. Phys. 229, 8095–8104 (2010).CrossRefGoogle Scholar
[57]Wang, H. and Basu, T., A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comp. 34, A2444-A2458 (2012).CrossRefGoogle Scholar
[58]Wang, Z., Vong, S. and Lei, S., Finite difference schemes for a two-dimensional time-space fractional differential equation, Inter. J. Comput. Math. (accepted for publication).Google Scholar
[59]Zaslavsky, G.M., Stevens, D. and Weitzner, H., Self-similar transport in incomplete chaos, Phys. Rev. E 48, 1683–1694 (1993).CrossRefGoogle ScholarPubMed
[60]Zhou, H., Tian, W. and Deng, W., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comp. 56, 45–66 (2013).CrossRefGoogle Scholar