369
Views
5
CrossRef citations to date
0
Altmetric
Original Articles

An approximate inverse preconditioner for spatial fractional diffusion equations with piecewise continuous coefficients

, &
Pages 523-545 | Received 28 Nov 2017, Accepted 11 Mar 2018, Published online: 14 Feb 2019

References

  • D. Benson, R. Schumer, M. Meerschaert, and S. Wheatcraft, Fractional dispersion, Lévy motion, and the MADE tracer tests, Transp. Porous Med. 42 (2001), pp. 211–240. doi: 10.1023/A:1006733002131
  • M. Benzi, Preconditioning techniques for large linear systems: A survey, J. Comput. Phys. 182 (2002), pp. 418–477. doi: 10.1006/jcph.2002.7176
  • R. Chan and G. Strang, Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Statist. Comput. 10 (1989), pp. 104–119. doi: 10.1137/0910009
  • M. Donatelli, M. Mazza, and S. Serra-Capizzano, Spectral analysis and structure preserving preconditioners for fractional diffusion equations, J. Comput. Phys. 307 (2016), pp. 262–279. doi: 10.1016/j.jcp.2015.11.061
  • R. Hilfer, Applications of Fractional Calculus in Physics, Word Scientific, Singapore, 2000.
  • S. Jaffard, Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), pp. 461–476. doi: 10.1016/S0294-1449(16)30287-6
  • J.W. Kirchner, X. Feng, and C. Neal, Fractal stream chemistry and its implications for contaminant transport in catchments, Nature 403 (2000), pp. 524–527. doi: 10.1038/35000537
  • S. Lei and H. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys. 242 (2013), pp. 715–725. doi: 10.1016/j.jcp.2013.02.025
  • X. Li and C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commum. Comput. Phys. 8 (2010), pp. 1016–1051.
  • F. Lin, S. Yang, and X. Jin, Preconditioned iterative methods for fractional diffusion equations, J. Comput. Phys. 256 (2014), pp. 109–117. doi: 10.1016/j.jcp.2013.07.040
  • X. Lin, M. Ng, and H. Sun, A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations, J. Comput. Phys. 336 (2017), pp. 69–86. doi: 10.1016/j.jcp.2017.02.008
  • X. Lin, M. Ng, and H. Sun, Stability and convergence analysis of finite difference schemes for time-dependent space-fractional diffusion equations with variable diffusion coefficients, J. Sci. Comput. 75 (2018), pp. 1102–1127. doi: 10.1007/s10915-017-0581-x
  • R.L. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, 2006.
  • M.M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), pp. 65–77. doi: 10.1016/j.cam.2004.01.033
  • M.M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), pp. 80–90. doi: 10.1016/j.apnum.2005.02.008
  • H. Moghaderi, M. Dehghan, M. Donatelli, and M. Mazza, Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations, J. Comput. Phys. 350 (2017), pp. 992–1011. doi: 10.1016/j.jcp.2017.08.064
  • M. Ng, Iterative Methods for Toeplitz Systems, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004.
  • J. Pan, R. Ke, M. Ng, and H. Sun, Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations, SIAM J. Sci. Comput. 36 (2014), pp. A2698–A2719. doi: 10.1137/130931795
  • J. Pan, M. Ng, and H. Wang, Fast iterative solvers for linear systems arising from time-dependent space-fractional diffusion equations, SIAM J. Sci. Comput. 38 (2016), pp. A2806–A2826. doi: 10.1137/15M1030273
  • H. Pang and H. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys. 231 (2012), pp. 693–703. doi: 10.1016/j.jcp.2011.10.005
  • H. Pang and H. Sun, Fast numerical Contour integral method for fractional diffusion equations, J. Sci. Comput. 66 (2016), pp. 41–66. doi: 10.1007/s10915-015-0012-9
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  • M. Raberto, E. Scalas, and F. Mainardi, Waiting-times and returns in high-frequency financial data: An empirical study, Physica A 314 (2002), pp. 749–755. doi: 10.1016/S0378-4371(02)01048-8
  • T. Strohmer, Four short stories about Toeplitz matrix calculations, Linear Algebra Appl. 343–344 (2002), pp. 321–344. doi: 10.1016/S0024-3795(01)00243-9
  • C. Tadjeran, M.M. Meerchaert, and H.P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), pp. 205–213. doi: 10.1016/j.jcp.2005.08.008
  • H. Wang and T.S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput. 34 (2012), pp. A2444–A2458. doi: 10.1137/12086491X
  • H. Wang and K. Wang, An O(Nlog2⁡N) alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. Comput. Phys. 230 (2011), pp. 7830–7839. doi: 10.1016/j.jcp.2011.07.003
  • H. Wang, K. Wang, and T. Sircar, A direct O(Nlog2⁡N) finite difference method for fractional diffusion equations, J. Comput. Phys. 229 (2010), pp. 8095–8104. doi: 10.1016/j.jcp.2010.07.011
  • L. Zhang, H. Sun, and H. Pang, Fast numerical solution for fractional diffusion equations by exponential quadrature rule, J. Comput. Phys. 299 (2015), pp. 130–143. doi: 10.1016/j.jcp.2015.07.001
  • H. Zhou, W. Tian, and W. Deng, Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput. 56 (2013), pp. 45–66. doi: 10.1007/s10915-012-9661-0

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.