Abstract
A modified Crank–Nicolson-type compact alternating direction implicit (ADI) finite difference method is proposed for a class of two-dimensional fractional subdiffusion equations with a time Riemann–Liouville fractional derivative of order \((1-\alpha )\) \((0<\alpha <1)\). This method improves the known compact ADI methods in the sense that it is based on the L1 approximation for the fractional derivative, the truncation errors on all time levels have the same order of \(\mathcal{O}(\tau ^{2\alpha }+h_{x}^{4}+h_{y}^{4})\) and the optimal error estimate \(\mathcal{O}(\tau ^{2\alpha }+h_{x}^{4}+h_{y}^{4})\) can be easily obtained in the standard \(H^{1}\)- and \(L^{2}\)-norms and the weighted \(L^{\infty }\)-norm. The unique solvability, unconditional stability and convergence of the resulting scheme are rigorously proved. A Richardson extrapolation algorithm is presented to increase the temporal accuracy from the order \(2\alpha \) to the order \(\min \{1+\alpha , 4\alpha \}\). Numerical results demonstrate the accuracy of the modified compact ADI method and the high efficiency of the Richardson extrapolation algorithm.
Similar content being viewed by others
References
Bouchaud, J., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)
Chen, A., Li, C.P.: A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions. Int. J. Comput. Math. (2015). doi:10.1080/00207160.1009905
Chen, C., Liu, F., Anh, V., Turner, I.: Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Comput. 81, 345–366 (2012)
Chen, C., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-diffusion equation. Appl. Math. Comput. 198, 754–769 (2008)
Chen, C., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886–897 (2007)
Chen, C., Liu, F., Turner, I., Anh, V.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32, 1740–1760 (2010)
Chen, C., Liu, F., Turner, I., Anh, V.: Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. Numer. Algorithms 54, 1–21 (2010)
Chen, S., Liu, F., Zhuang, P., Anh, V.: Finite difference approximations for the fractional Fokker–Planck equation. Appl. Math. Model. 33, 256–273 (2009)
Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)
Gao, G.H., Sun, Z.Z.: A compact difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)
Giona, M., Roman, H.E.: Fractional diffusion equation for transport phenomena in random media. Phys. A 185, 87–97 (1992)
Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284, 521–541 (2002)
Gu, Y., Zhuang, P., Liu, F.: An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. Comput. Model. Eng. Sci. 56, 303–334 (2010)
Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Phys. A 276, 448–455 (2000)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Kosztolowicz, T.: Subdiffusion in a system with a thick membrane. J. Membr. Sci. 320, 492–499 (2008)
Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)
Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)
Liao, H.L., Sun, Z.Z., Shi, H.S.: Error estimate of fourth-order compact scheme for linear Schrödinger equations. SIAM J. Numer. Anal. 47, 4381–4401 (2010)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
Liu, Q., Gu, Y.T., Zhuang, P., Liu, F., Nie, Y.F.: An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48, 1–12 (2011)
Liu, Q., Liu, F., Turner, I., Anh, V.: Finite element approximation for a modified anomalous subdiffusion equation. Appl. Math. Model. 35, 4103–4116 (2011)
Liu, F., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231, 160–176 (2009)
Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker–Planck equation approach. Phys. Rev. Lett. 82, 3563–3567 (1999)
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Phys. A 278, 107–125 (2000)
Mohebbi, A., Abbaszade, M., Dehghan, M.: A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term. J. Comput. Phys. 240, 36–48 (2013)
Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York (2001)
Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)
Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)
Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von-Neumann type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)
Zeng, F.H., Li, C.P., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, A2976–A3000 (2013)
Zeng, F.H., Li, C.P., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37, A55–A78 (2015)
Zhang, Y.N., Sun, Z.Z.: Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation. J. Sci. Comput. 59, 104–128 (2014)
Zhang, Y.N., Sun, Z.Z., Wu, H.W.: Error estimates of Crank–Nicolson-type difference schemes for the subdiffusion equation. SIAM J. Numer. Anal. 49, 2302–2322 (2011)
Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)
Zhuang, P., Liu, F.: Finite difference approximation for two-dimensional time fractional diffusion equation. J. Algorithms Comput. Technol. 1, 1–15 (2007)
Zhuang, P., Liu, F., Anh, V., Turner, I.: Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J. Appl. Math. 74, 645–667 (2009)
Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)
Acknowledgments
The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper. This work was supported in part by E-Institutes of Shanghai Municipal Education Commission No. E03004, Science and Technology Commission of Shanghai Municipality (STCSM) No. 13dz2260400, and Shanghai Leading Academic Discipline Project No. B407.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, T., Wang, YM. A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations. J. Appl. Math. Comput. 52, 439–476 (2016). https://doi.org/10.1007/s12190-015-0949-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-015-0949-8
Keywords
- Fractional subdiffusion equation
- Compact ADI method
- Finite difference scheme
- Error estimate
- Richardson extrapolation