Calcolo

, Volume 53, Issue 3, pp 301–330 | Cite as

Error analysis of a high-order compact ADI method for two-dimensional fractional convection-subdiffusion equations

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Abstract

This paper is concerned with numerical methods for a class of two-dimensional fractional convection-subdiffusion equations with a time Caputo fractional derivative of order \(\alpha (0<\alpha <1)\). We first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation in the spatial directions and by an alternating direction implicit (ADI) approximation in the temporal direction. The resulting compact ADI scheme is uniquely solvable and unconditionally stable. The optimal error estimates in the weighted \(L^{\infty }\), \(H^{1}\) and \(L^{2}\) norms are obtained, and show that the compact ADI method has the temporal accuracy of order \(\min \{1+\alpha ,2-\alpha \}\) and the fourth-order spatial accuracy. Applications using three model problems give numerical results that demonstrate the accuracy and the effectiveness of this new method.

Keywords

Fractional convection-subdiffusion equation Compact ADI method Finite difference scheme Stability and convergence Error estimate 

Mathematics Subject Classification

65M06 65M12 65M15 35R11 
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Notes

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper.

References

  1. 1.
    Bouchaud, J., Georses, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Solomon, T.H., Weeks, E.R., Swiney, H.L.: Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow. Phys. Rev. Lett. 71, 3975–3979 (1993)CrossRefGoogle Scholar
  3. 3.
    Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)MATHGoogle Scholar
  4. 4.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)CrossRefMATHGoogle Scholar
  5. 5.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)MATHGoogle Scholar
  7. 7.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic, New York (1974)MATHGoogle Scholar
  8. 8.
    Fujita, Y.: Integrodifferential equation which interpolates the heat equation and the wave equation, I. Osaka J. Math. 27, 309–321 (1990)MathSciNetMATHGoogle Scholar
  9. 9.
    Fujita, Y.: Integrodifferential equation which interpolates the heat equation and the wave equation, II. Osaka J. Math. 27, 797–804 (1990)MathSciNetMATHGoogle Scholar
  10. 10.
    Klages, R., Radons, G., Sokolov, I.M.: Anomalous Transport: Foundations and Applications. Wiley-VCH, Weinheim (2008)CrossRefGoogle Scholar
  11. 11.
    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)CrossRefMATHGoogle Scholar
  12. 12.
    Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. Math. Gen. 37, R161–R208 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129–143 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wyss, W.: Fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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)MathSciNetMATHGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gao, G.H., Sun, Z.Z.: A compact difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Burrage, K., Hale, N., Kay, D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34, A2145–A2172 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855–875 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhen, Y.Y., Li, C.P., Zhao, Z.G.: A note on the finite element method for the space-fractional advection diffusion equation. Comput. Math. Appl. 59, 1718–1726 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT. doi: 10.1007/s10543-014-0484-2
  32. 32.
    Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    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)MathSciNetMATHGoogle Scholar
  35. 35.
    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)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    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)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zhuang, P., Liu, F.: Finite difference approximation for two-dimensional time fractional diffusion equation. J. Algorithms Comput. Technol. 1, 1–15 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229, 6613–6622 (2010)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Yang, Q., Moroney, T., Burrage, K., Turner, I., Liu, F.: Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions. ANZIAM J. Electron. Suppl. 52, C395–C409 (2010)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Yang, Q., Turner, I., Liu, F., Ilić, M.M.: Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions. SIAM J. Sci. Comput. 33, 1159–1180 (2011)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zhang, Y.N., Sun, Z.Z.: Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 230, 8713–8728 (2011)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Deng, W.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Cui, M.: Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer. Algorithms 62, 383–409 (2013)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    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)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Cui, M.: Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J. Comput. Phys. 231, 2621–2633 (2012)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Cui, M.: A high-order compact exponential scheme for the fractional convection-diffusion equation. J. Comput. Appl. Math. 255, 404–416 (2014)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Mohebbi, A., Abbaszadeh, M.: Compact finite difference scheme for the solution of time fractional advection-dispersion equation. Numer. Algorithms 63, 431–452 (2013)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Zhao, X., Xu, Q.: Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient. Appl. Math. Model. 38, 3848–3859 (2014)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zhao, X., Sun, Z.Z.: Compact Crank–Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62, 747–771 (2015)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Chen, S., Liu, F., Zhuang, P., Anh, V.: Finite difference approximations for the fractional Fokker–Planck equation. Appl. Math. Model. 33, 256–273 (2009)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Sun, Z.Z.: The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations. Science Press, Beijing (2009)Google Scholar
  53. 53.
    Liao, H.L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods. Partial Differ. Equ. 26, 37–60 (2010)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Zhao, X., Sun, Z.Z., Karniadakis, G.E.: Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293, 184–200 (2015)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Dover Publications Inc., New York (1994)MATHGoogle Scholar
  56. 56.
    Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York (2001)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghaiPeople’s Republic of China
  2. 2.Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science, E-Institute of Shanghai UniversitiesShanghai Normal UniversityShanghaiPeople’s Republic of China

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