Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T05:24:25.272Z Has data issue: false hasContentIssue false
  • English
  • Français

Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations

Published online by Cambridge University Press:  30 July 2015

Xiao-Qing Jin ,
Fu-Rong Lin  and
Zhi Zhao
Xiao-Qing Jin
Affiliation:
Department of Mathematics, University of Macau, Macao 999078, China
Fu-Rong Lin
Affiliation:
Department of Mathematics, Shantou University, Shantou 515063, China
Zhi Zhao*
Affiliation:
Department of Mathematics, University of Macau, Macao 999078, China
*
*Corresponding author. Email addresses: [email protected] (X.-Q. Jin), [email protected] (F.-R. Lin), [email protected] (Z. Zhao)
Get access

Abstract

In this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou and W. Deng, A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretized linear systems, we first propose preconditioned iterative methods to solve them. Then we apply the D’Yakonov ADI scheme to split the linear systems and solve the obtained splitting systems by iterative methods. Two preconditioned iterative methods, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. By fully exploiting the structure of the coefficient matrix, we design two special kinds of preconditioners, which are easily constructed and are able to accelerate convergence of iterative solvers. Numerical results show the efficiency of our preconditioners.

Keywords

65F1065L0665U05Fractional diffusion equationCN-WSGD schemepreconditioned GMRES methodpreconditioned CGNR methodToeplitz matrixfast Fourier transform
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 2 , August 2015 , pp. 469 - 488
Copyright
Copyright © Global-Science Press 2015 

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]Axelsson, O., Iterative Solution Methods, Cambridge University Press, 1996.Google Scholar
[2]Bai, J. and Feng, X., Fractional-order anisotropic diffusion for image denoising, IEEE Tran. Image Process., 16 (2007) 2492–2502.Google Scholar
[3]Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J. M., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. and der Vorst, H. V., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.CrossRefGoogle Scholar
[4]Basu, T. S. and Wang, H., A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Model., 9 (2012) 658–666.Google Scholar
[5]Benson, D., Wheatcraft, S. W. and Meerchaert, M. M., Applications of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000) 1403–1413.Google Scholar
[6]Benson, D., Wheatcraft, S. W. and Meerschaert, M. M., The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000) 1413–1423.CrossRefGoogle Scholar
[7]Benson, D., Kovács, M. and Meerchaert, M. M., Numerical solutions for fractional reaction-diffusion equations, Comput. Math. Appl., 55 (2008) 2212–2226.Google Scholar
[8]Carreras, B. A., Lynch, V. E. and Zaslavsky, G. M., Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8 (2001) 5096–5103.CrossRefGoogle Scholar
[9]Chan, R., Chang, Q. and Sun, H., Multigrid method for ill-conditioned symmetric Toeplitz systems, SIAM J. Sci. Statist. Comput., 19 (1998) 516–529.Google Scholar
[10]Chan, R. and Jin, X., An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.Google Scholar
[11]Chan, R. and Ng, M., Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996) 427–482.Google Scholar
[12]Chen, M. and Deng, W., Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators, Commun. Comput. Phys., 16 (2014) 516–540.Google Scholar
[13]Deng, W. and Chen, M., Efficient numerical algorithms for three-dimensional fractional partial differential equations, J. Comp. Math., 32 (2014) 371–391.Google Scholar
[14]Ervin, V. J., Heuer, N. and Roop, J. P., Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45 (2007) 572–591.CrossRefGoogle Scholar
[15]Ford, N. J. and Simpson, A. C., The numerical solution of fractional differential equations: speed versus accuracy, Numer. Algorithms, 26 (2001) 333–346.Google Scholar
[16]Ji, X. and Tang, H., High-order accurate Runge-Kutta (local) discontinuous Galerkin methods for one- and two-dimensional fractional diffusion equations, Numer. Math. Theor. Meth. Appl., 5 (2012) 333–358.Google Scholar
[17]Jin, X., Preconditioning Techniques for Toeplitz Systems, Higher Education Press, Beijing, 2010.Google Scholar
[18]Langlands, T. A. M., Henry, B. I. and Wearne, S. L., Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions, J. Math. Biol., 59 (2009) 761–808.Google Scholar
[19]Lei, S. and Sun, H., A circulant preconditioner for fractional diffusion equation, J. Comput. Phys., 242 (2013) 715–725.Google Scholar
[20]Lin, F., Yang, S. and Jin, X., Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256 (2014) 109–117.CrossRefGoogle Scholar
[21]Liu, F., Anh, V. and Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2004) 209–219.Google Scholar
[22]Magin, R. L., Fractional Calculus in Bioengineering, Begell House Publishers, 2006.Google Scholar
[23]Mainardi, F., Fractals and fractional calculus continuum mechanics, Springer Verlag, 1997, pp. 291–348.Google Scholar
[24]Meerschaert, M. M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Phys., 172 (2004) 65–77.Google Scholar
[25]Meerschaert, M. M. and Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006) 80–90.CrossRefGoogle Scholar
[26]Meerschaert, M. M., Scheffler, H. P. and Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006) 249–261.Google Scholar
[27]Murio, D. A., Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008) 1138–1145.Google Scholar
[28]Oldham, K. B., Fractional differential equations in electrochemistry, Adv. Engrg. Software, 41 (2010) 9–12 (Civil-Comp Special Issue).Google Scholar
[29]Pang, H. and Sun, H., Multigrid iterative methods for fractional diffusion equations, J. Comput. Phys., 231 (2012) 693–703.Google Scholar
[30]Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.Google Scholar
[31]Raberto, M., Scalas, E. and Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A, 314 (2002) 749–755.CrossRefGoogle Scholar
[32]Ren, J. and Sun, Z., Efficient and stable numerical methods for multi-term time fractional sub-diffusion equations, East Asian Journal on Applied Mathematics, 4 (2014) 242–266.Google Scholar
[33]Ritchie, K., Shan, X., 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 (2005) 2266–2277.Google Scholar
[34]Roop, J. P., Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2, J. Comput.Appl. Math., 193 (2006) 243–268.Google Scholar
[35]Scher, H. and Montroll, E. W., Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12 (1975) 2455–2477.Google Scholar
[36]Sokolov, I. M., Klafter, J. and Blumen, A., Fractional kinetics, Phys. Today (Nov. 2002), pp. 48–54.Google Scholar
[37]Sousa, E., Finite difference approximates for a fractional advection diffusion problem, J. Comput. Phys., 228 (2009) 4038–4054.Google Scholar
[38]Tadjeran, C., Meerschaert, M. M. and Scheffler, H. P., A second-order accurate numerical approximation for the fractional diffusion equations, J. Comput. Phys., 213 (2006) 205–213.Google Scholar
[39]Tian, W., Zhou, H. and Deng, W., A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703–1727.Google Scholar
[40]Wang, H. and Wang, K., An O(Nlog2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. Comput. Phys., 230 (2011) 7830–7839.Google Scholar
[41]Wang, H., Wang, K. and Sircar, T., A direct O(Nlog2N) finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010) 8095–8104.Google Scholar
[42]Wang, K. and Wang, H., A fast characteristic finite difference method for fractional advection-diffusion equations, Adv Water Resour., 34 (2011) 810–816.Google Scholar
[43]Wang, H. and Basu, T. S., A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012) 2444–2458.Google Scholar
[44]Zhang, N., Deng, W. and Wu, Y., Finite difference/element method for a two-dimensional modified fractional diffusion equation, Adv. Appl. Math. Mech., 4 (2012) 496–518.Google Scholar