Abstract
In this paper, we study and analyze Crank–Nicolson temporal discretization with high-order spatial difference schemes for time-dependent Riesz space-fractional diffusion equations with variable diffusion coefficients. To the best of our knowledge, there is no stability and convergence analysis for temporally 2nd-order or spatially jth-order (\(j\ge 3\)) difference schemes for such equations with variable coefficients. We prove under mild assumptions on diffusion coefficients and spatial discretization schemes that the resulting discretized systems are unconditionally stable and convergent with respect to discrete \(\ell ^2\)-norm. We further show that several spatial difference schemes with jth-order (\(j=1,2,3,4\)) truncation error satisfy the assumptions required in our analysis. As a result, we obtain a series of temporally 2nd-order and spatially jth-order (\(j=1,2,3,4\)) unconditionally stable difference schemes for solving time-dependent Riesz space-fractional diffusion equations with variable coefficients. Numerical results are presented to illustrate our theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agrawal, O.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 29, 145–155 (2002)
Benson, D., Wheatcraft, S., Meerschaert, M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)
Bouchaud, J., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)
Çelik, C., Duman, M.: Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)
Chen, M., Deng, W., Wu, Y.: Superlinearly convergent algorithms for the two-dimensional space time Caputo Riesz fractional diffusion equation. Appl. Numer. Math. 70, 22–41 (2013)
Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014)
del Castillo-Negrete, D., Carreras, B., Lynch, V.: Fractional diffusion in Plasma turbulence. Phys. Plasmas 11, 3854–3864 (2004)
Ding, H., Li, C.: High-order numerical algorithms for Riesz derivatives via constructing new generating functions. J. Sci. Comput. 71, 759–784 (2017)
Hao, Z., Sun, Z., Cao, W.: A fourth-order approximation of fractional derivatives with its applications. J. Comput. Phys. 281, 787–805 (2015)
Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Lei, S., Huang, Y.: Fast algorithms for high-order numerical methods for space-fractional diffusion equations. Int. J. Comput. Math. 94, 1062–1078 (2017)
Lin, X., Ng, M., Sun, H.: A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations. J. Comput. Phys. 336, 69–86 (2017)
Liu, Q., Liu, F., Gu, Y., Zhuang, P., Chen, J., Turner, I.: A meshless method based on point interpolation method (PIM) for the space fractional diffusion equation. Appl. Math. Comput. 256, 930–938 (2015)
Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equation. J. Comput. Appl. Math. 172, 65–77 (2004)
Meerschaert, M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
Ng, M.: Iterative Methods for Toeplitz Systems. Oxford University Press, Oxford (2004)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Solomon, T., Weeks, E., Swinney, H.: Observation of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow. Phys. Rev. 71, 3975–3979 (1993)
Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann Liouville derivative. Appl. Numer. Math. 90, 22–37 (2015)
Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comp. 84, 1703–1727 (2015)
Wang, H., Wang, K., Sircar, T.: A direct \(O(N\log ^2 N)\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)
Yang, Z., Yuan, Z., Nie, Y., Wang, J., Zhu, X., Liu, F.: Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains. J. Comput. Phys. 330, 863–883 (2017)
Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I., Anh, V.: A Crank–Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52, 2599–2622 (2014)
Zhang, Y., Ding, H.: High-order algorithm for the two-dimension Riesz space-fractional diffusion equation. Int. J. Comput. Math. 94, 2063–2073 (2017)
Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by HKRGC GRF 12200317, 12306616, 12302715, 12301214.
Research supported by MYRG2016-00063-FST from University of Macau and 054/2015/A2 from FDCT of Macao.
Rights and permissions
About this article
Cite this article
Lin, Xl., Ng, M.K. & Sun, HW. Stability and Convergence Analysis of Finite Difference Schemes for Time-Dependent Space-Fractional Diffusion Equations with Variable Diffusion Coefficients. J Sci Comput 75, 1102–1127 (2018). https://doi.org/10.1007/s10915-017-0581-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-017-0581-x
Keywords
- Time-dependent space-fractional diffusion equation
- Variable diffusion coefficients
- High-order finite difference schemes
- Stability
- Convergence