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Volumn 62, Issue 3, 2013, Pages 383-409

Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation

Author keywords

Alternating direction implicit; Compact scheme; Convergence; Finite difference; Multidimensional fractional differential equation; Stability

Indexed keywords


EID: 84874394694     PISSN: 10171398     EISSN: 15729265     Source Type: Journal    
DOI: 10.1007/s11075-012-9589-3     Document Type: Article
Times cited : (82)

References (39)
  • 1
    • 77954142858 scopus 로고    scopus 로고
    • Numerical simulations of 2D fractional subdiffusion problems
    • Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229, 6613-6622 (2010).
    • (2010) J. Comput. Phys. , vol.229 , pp. 6613-6622
    • Brunner, H.1    Ling, L.2    Yamamoto, M.3
  • 2
    • 36149001420 scopus 로고    scopus 로고
    • A Fourier method for the fractional diffusion equation describing sub-diffusion
    • Chen, C. M., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886-897 (2007).
    • (2007) J. Comput. Phys. , vol.227 , pp. 886-897
    • Chen, C.M.1    Liu, F.2    Turner, I.3    Anh, V.4
  • 3
    • 51749116733 scopus 로고    scopus 로고
    • Finite difference approximations for the fractional Fokker-Plank equation
    • Chen, S., Liu, F., Zhuang, P., Anh, V.: Finite difference approximations for the fractional Fokker-Plank equation. Appl. Math. Model. 33, 256-273 (2009).
    • (2009) Appl. Math. Model. , vol.33 , pp. 256-273
    • Chen, S.1    Liu, F.2    Zhuang, P.3    Anh, V.4
  • 4
    • 77950690888 scopus 로고    scopus 로고
    • Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation
    • Chen, C. M., 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).
    • (2010) Numer. Algorithms , vol.54 , pp. 1-21
    • Chen, C.M.1    Liu, F.2    Turner, I.3    Anh, V.4
  • 5
    • 69049086472 scopus 로고    scopus 로고
    • Compact finite difference method for the fractional diffusion equation
    • Cui, M. R.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792-7804 (2009).
    • (2009) J. Comput. Phys. , vol.228 , pp. 7792-7804
    • Cui, M.R.1
  • 6
    • 84856219458 scopus 로고    scopus 로고
    • Compact alternating direction implicit method for two-dimensional time fractional diffusion equation
    • Cui, M. R.: Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J. Comput. Phys. 231, 2621-2633 (2012).
    • (2012) J. Comput. Phys. , vol.231 , pp. 2621-2633
    • Cui, M.R.1
  • 7
    • 0012728672 scopus 로고
    • Alternating direction methods for nonlinear time-dependent problems
    • Dendy, J. E. Jr.: Alternating direction methods for nonlinear time-dependent problems. SIAM J. Numer. Anal. 14, 313-326 (1977).
    • (1977) SIAM J. Numer. Anal. , vol.14 , pp. 313-326
    • Dendy, J.E.1    Dendy Jr., J.E.2
  • 9
    • 0001742634 scopus 로고
    • A general formulation of alternating direction method, part I. Parabolic and hyperbolic problems
    • Douglas, J. Jr., Gunn, J. E.: A general formulation of alternating direction method, part I. Parabolic and hyperbolic problems. Numer. Math. 6, 428-453 (1964).
    • (1964) Numer. Math. , vol.6 , pp. 428-453
    • Douglas Jr., J.1    Gunn, J.E.2
  • 10
    • 42649109055 scopus 로고    scopus 로고
    • Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation
    • Ervin, V. J., Heuer, N., Roop, J. P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572-591 (2007).
    • (2007) SIAM J. Numer. Anal. , vol.45 , pp. 572-591
    • Ervin, V.J.1    Heuer, N.2    Roop, J.P.3
  • 11
    • 78649334165 scopus 로고    scopus 로고
    • A compact finite difference scheme for the fractional sub-diffusion equations
    • Gao, G. H., Sun, Z. Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586-595 (2011).
    • (2011) J. Comput. Phys. , vol.230 , pp. 586-595
    • Gao, G.H.1    Sun, Z.Z.2
  • 12
    • 0003102291 scopus 로고
    • Higher order accurate difference solutions of fluid mechanics problems by a compact difference technique
    • Hirsch, R. S.: Higher order accurate difference solutions of fluid mechanics problems by a compact difference technique. J. Comput. Phys. 24, 90-109 (1975).
    • (1975) J. Comput. Phys. , vol.24 , pp. 90-109
    • Hirsch, R.S.1
  • 13
    • 17144427014 scopus 로고    scopus 로고
    • The accuracy and stability of an implicit solution method for the fractional diffusion equation
    • 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).
    • (2005) J. Comput. Phys. , vol.205 , pp. 719-736
    • Langlands, T.A.M.1    Henry, B.I.2
  • 14
    • 9144220381 scopus 로고
    • Compact finite difference schemes with spectral-like resolution
    • Lele, S. K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16-42 (1992).
    • (1992) J. Comput. Phys. , vol.103 , pp. 16-42
    • Lele, S.K.1
  • 15
    • 79952454978 scopus 로고    scopus 로고
    • Numerical approaches to fractional calculus and fractional ordinary differential equation
    • Li, C. P., Chen, A., Ye, J. J.: Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230, 3352-3368 (2011).
    • (2011) J. Comput. Phys. , vol.230 , pp. 3352-3368
    • Li, C.P.1    Chen, A.2    Ye, J.J.3
  • 17
    • 77958536189 scopus 로고    scopus 로고
    • A space-time spectral method for the time fractional diffusion equation
    • Li, X., Xu, Ch.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108-2131 (2009).
    • (2009) SIAM J. Numer. Anal. , vol.47 , pp. 2108-2131
    • Li, X.1    Xu, C.2
  • 18
    • 73449143894 scopus 로고    scopus 로고
    • Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations
    • 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).
    • (2010) Numer. Methods Partial Differ. Equ. , vol.26 , pp. 37-60
    • Liao, H.L.1    Sun, Z.Z.2
  • 19
    • 0036569404 scopus 로고    scopus 로고
    • An efficient high-order algorithm for solving systems of reaction-diffusion equations
    • Liao, W., Zhu, J., Khaliq, A. Q. M.: An efficient high-order algorithm for solving systems of reaction-diffusion equations. Numer. Methods Partial Differ. Equ. 18, 340-354 (2002).
    • (2002) Numer. Methods Partial Differ. Equ. , vol.18 , pp. 340-354
    • Liao, W.1    Zhu, J.2    Khaliq, A.Q.M.3
  • 20
    • 25444463578 scopus 로고    scopus 로고
    • Finite difference methods for two-dimensional fractional dispersion equation
    • Meerschaert, M. M., Scheffler, H. P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211, 249-261 (2006).
    • (2006) J. Comput. Phys. , vol.211 , pp. 249-261
    • Meerschaert, M.M.1    Scheffler, H.P.2    Tadjeran, C.3
  • 21
    • 0002641421 scopus 로고    scopus 로고
    • The random walk's guide to anomalous diffusion: a fractional dynamics approach
    • Metzler, R., Klafter, J.: The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1-77 (2000).
    • (2000) Phys. Rep. , vol.339 , pp. 1-77
    • Metzler, R.1    Klafter, J.2
  • 24
    • 0002058827 scopus 로고
    • The numerical solution of parabolic and elliptic differential equations
    • Peaceman, D. W., Rachford, H. H. Jr.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28-41 (1955).
    • (1955) J. Soc. Ind. Appl. Math. , vol.3 , pp. 28-41
    • Peaceman, D.W.1    Rachford Jr., H.H.2
  • 26
    • 33744801528 scopus 로고    scopus 로고
    • Matrix approach to discrete fractional calculus
    • Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 3, 359-386 (2000).
    • (2000) Fract. Calc. Appl. Anal. , vol.3 , pp. 359-386
    • Podlubny, I.1
  • 27
    • 61349186917 scopus 로고    scopus 로고
    • Matrix approach to discrete fractional calculus II: partial fractional differential equations
    • Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., Jara, B. M. V.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228, 3137-3153 (2009).
    • (2009) J. Comput. Phys. , vol.228 , pp. 3137-3153
    • Podlubny, I.1    Chechkin, A.2    Skovranek, T.3    Chen, Y.4    Jara, B.M.V.5
  • 28
    • 0004029439 scopus 로고    scopus 로고
    • Implicit, compact, linearized θ-methods with factorization for multidimensional reaction-diffusion equations
    • Ramos, J. I.: Implicit, compact, linearized θ-methods with factorization for multidimensional reaction-diffusion equations. Appl. Math. Comput. 94, 17-43 (1998).
    • (1998) Appl. Math. Comput. , vol.94 , pp. 17-43
    • Ramos, J.I.1
  • 30
    • 79951851714 scopus 로고    scopus 로고
    • Numerical approximations and solution techniques for the space-time RieszCCaputo fractional advection-diffusion equation
    • Shen, S., Liu, F., Anh, V.: Numerical approximations and solution techniques for the space-time RieszCCaputo fractional advection-diffusion equation. Numer. Algorithms 56, 383-403 (2011).
    • (2011) Numer. Algorithms , vol.56 , pp. 383-403
    • Shen, S.1    Liu, F.2    Anh, V.3
  • 31
    • 54549108683 scopus 로고    scopus 로고
    • A compact alternating direction implicit difference method for reaction diffusion equations
    • (in Chinese)
    • Sun, Z. Z., Li, X. L.: A compact alternating direction implicit difference method for reaction diffusion equations. Math. Numer. Sin. 27, 209-224 (2005) (in Chinese).
    • (2005) Math. Numer. Sin. , vol.27 , pp. 209-224
    • Sun, Z.Z.1    Li, X.L.2
  • 32
    • 33845628108 scopus 로고    scopus 로고
    • A second-order accurate numerical method for the two-dimensional fractional diffusion equation
    • Tadjeran, C., Meerschaert, M. M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220, 813-823 (2007).
    • (2007) J. Comput. Phys. , vol.220 , pp. 813-823
    • Tadjeran, C.1    Meerschaert, M.M.2
  • 33
    • 0003423678 scopus 로고
    • Numerical partial differential equations: finite difference methods
    • Berlin: Springer
    • Thomas, J. W.: Numerical partial differential equations: finite difference methods. In: Texts in Applied Mathematics, vol. 22. Springer, Berlin (1995).
    • (1995) Texts in Applied Mathematics, Vol. 22
    • Thomas, J.W.1
  • 34
    • 77955927812 scopus 로고    scopus 로고
    • 2N) finite difference method for fractional diffusion equations
    • 2N) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095-8104 (2010).
    • (2010) J. Comput. Phys. , vol.229 , pp. 8095-8104
    • Wang, H.1    Wang, K.2    Sircar, T.3
  • 35
    • 84860390012 scopus 로고    scopus 로고
    • 2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations
    • 2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations. J. Comput. Phys. 230, 7830-7839 (2011).
    • (2011) J. Comput. Phys. , vol.230 , pp. 7830-7839
    • Wang, H.1    Wang, K.2
  • 37
    • 25444472344 scopus 로고    scopus 로고
    • An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations
    • 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).
    • (2005) SIAM J. Numer. Anal. , vol.42 , pp. 1862-1874
    • Yuste, S.B.1    Acedo, L.2
  • 38
    • 80053633596 scopus 로고    scopus 로고
    • Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation
    • 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).
    • (2011) J. Comput. Phys. , vol.230 , pp. 8713-8728
    • Zhang, Y.N.1    Sun, Z.Z.2
  • 39
    • 84907893973 scopus 로고    scopus 로고
    • Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term
    • 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).
    • (2009) SIAM J. Numer. Anal. , vol.47 , pp. 1760-1781
    • Zhuang, P.1    Liu, F.2    Anh, V.3    Turner, I.4


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