418
Views
4
CrossRef citations to date
0
Altmetric
Articles

A circulant preconditioner for the Riesz distributed-order space-fractional diffusion equations

, , ORCID Icon &
Pages 3081-3096 | Received 22 Feb 2020, Accepted 08 Sep 2020, Published online: 24 Sep 2020

References

  • Caputo M. Mean fractional-order-derivatives differential equations and filters. Annali dell'Universita Di Ferrara. 1995;41:73–84.
  • Caputo M. Distributed order differential equations modelling dielectric induction and diffusion. Fract Calc Appl Anal. 2001;4:421–442.
  • Caputo M. Diffusion with space memory modelled with distributed order space fractional differential equations. Ann Geophys. 2003;46:223–234.
  • Chechkin AV, Gorenflo R, Sokolov IM. Retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations. Phys Rev E. 2002;66:046129. doi: 10.1103/PhysRevE.66.046129
  • Kochubei AN. Distributed order calculus and equations of ultraslow diffusion. J Math Anal Appl. 2008;340:252–281. doi: 10.1016/j.jmaa.2007.08.024
  • Su N, Nelson PN, Connor S. The distributed-order fractional diffusion-wave equation of groundwater flow: theory and application to pumping and slug tests. J Hydrol. 2015;529:1262–1273. doi: 10.1016/j.jhydrol.2015.09.033
  • Fan W, Liu F. A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain. Appl Math Lett. 2018;77:114–121. doi: 10.1016/j.aml.2017.10.005
  • Li J, Liu F, Feng L, et al. A novel finite volume method for the Riesz space distributed-order diffusion equation. Comput Math Appl. 2017;74:772–783. doi: 10.1016/j.camwa.2017.05.017
  • Abbaszadeh M. Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation. Appl Math Lett. 2019;88:179–185. doi: 10.1016/j.aml.2018.08.024
  • Jia J, Wang H. A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains. Comput Math Appl. 2018;75:2031–2043. doi: 10.1016/j.camwa.2017.09.003
  • Zheng X, Huan L, Wang H, et al. An efficient finite volume method for nonlinear distributed-order space-fractional diffusion equations in three space dimensions. J Sci Comput.. 2019;80:1395–1418. doi: 10.1007/s10915-019-00979-2
  • Bu W, Liu X, Tang Y, et al. Finite element multigrid method for multi-term time fractional advection–diffusion equations. Int J Model Simul Sci Comput. 2015;6:1540001. doi: 10.1142/S1793962315400012
  • Donatelli M, Mazza M, Serra-Capizzano S. Spectral analysis and structure preserving preconditioners for fractional diffusion equation. J Comput Phys. 2016;307:262–279. doi: 10.1016/j.jcp.2015.11.061
  • Donatelli M, Mazza M, Serra-Capizzano S. Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations. SIAM J Sci Comput. 2018;40:A4007–A4039. doi: 10.1137/17M115164X
  • Lei S, Sun H. A circulant preconditioner for fractional diffusion equations. J Comput Phys. 2013;242:715–725. doi: 10.1016/j.jcp.2013.02.025
  • Mognaderi H, Dehghan M, Donatelli M, et al. Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations. J Comput Phys. 2017;350:992–1011. doi: 10.1016/j.jcp.2017.08.064
  • Pang H, Sun H. Multigrid method for fractional diffusion equations. J Comput Phys. 2012;231:693–703. doi: 10.1016/j.jcp.2011.10.005
  • Parvizi M, Eslahchi M, Dehghan M. Numerical solution of fractional advection–diffusion equation with a nonlinear source term. Numer Algorithms. 2015;68:601–629. doi: 10.1007/s11075-014-9863-7
  • Chan R, Ng M. Conjugate gradient methods for Toeplitz systems. SIAM Rev. 1996;38:427–482. doi: 10.1137/S0036144594276474
  • Jin X. Preconditioning techniques for Toeplitz systems. Beijing: Higher Education Press; 2010.
  • Chan R, Strang G. Toeplitz equations by conjugate gradients with circulant preconditioner. SIAM J Sci Statist Comput. 1989;10:104–119. doi: 10.1137/0910009
  • Hao Z, Sun Z, Cao W. A fourth-order approximation of fractional derivatives with its applications. J Comput Phys. 2015;281:787–805. doi: 10.1016/j.jcp.2014.10.053
  • Varga R. Matrix iterative analysis. New York: Springer; 1991.
  • Horn R, Johnson C. Matrix analysis. 2nd ed. Cambridge: Cambridge University Press; 2013.
  • Wang H, Wang K, Sircar T. A O(Nlog2⁡N) direct finite difference method for fractional diffusion equations. J Comput Phys.. 2010;229:4038–4054.
  • Serra S. Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems. Math Comput. 1997;66:651–665. doi: 10.1090/S0025-5718-97-00833-8
  • Serra S. Superlinear PCG methods for symmetric Toeplitz systems. Math Comput. 1999;68:793–803. doi: 10.1090/S0025-5718-99-01045-5

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.