zbMATH — the first resource for mathematics

Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. (English) Zbl 1242.65158
Summary: High-order compact finite difference scheme with operator splitting technique for solving two-dimensional time fractional diffusion equation is considered in this paper. A Grünwald-Letnikov approximation is used for the Riemann-Liouville time derivative, and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme. Alternating direction implicit (ADI) method is used to split the original problem into two separate one-dimensional problems. The local truncation error is analyzed and the stability is discussed by the Fourier method. The proposed scheme is suitable when the order of the time fractional derivative \(\gamma \) lies in the interval \(\gamma \in (0,\frac 12)\). A correction term is added to maintain high accuracy when \([\frac 12,1)\). Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
ma2dfc; PDE2D
Full Text: DOI
[1] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. rep., 339, 1-77, (2000) · Zbl 0984.82032
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[3] Lynch, V.E.; Carreras, B.A.; Del-Castillo-Negrete, D.; Ferreira-Mejias, K.M.; Hicks, H.R., Numerical methods for the solution of partial differential equations of fractional order, J. comput. phys., 192, 406-421, (2003) · Zbl 1047.76075
[4] Meerschaert, M.; Tadjeran, C., Finite difference approximations for fractional advection – dispersion flow equations, J. comput. appl. math., 172, 65-77, (2004) · Zbl 1126.76346
[5] 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) · Zbl 1072.65123
[6] 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, 5, 1862-1874, (2005) · Zbl 1119.65379
[7] Yuste, S.B., Weighted average finite difference methods for fractional diffusion equations, J. comput. phys., 216, 264-274, (2006) · Zbl 1094.65085
[8] 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, 2, 1079-1095, (2008) · Zbl 1173.26006
[9] Chen, C.-M.; Liu, F.; Burrage, K., Finite difference methods and a Fourier analysis for the fractional reaction – subdiffusion equation, Appl. math. comput., 198, 754-769, (2008) · Zbl 1144.65057
[10] Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y.; Vinagre Jara, B.M., Matrix approach to discrete fractional calculus II: partial fractional differential equations, J. comput. phys., 228, 3137-3153, (2009) · Zbl 1160.65308
[11] 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) · Zbl 1170.65107
[12] Du, R.; Cao, W.R.; Sun, Z.Z., A compact difference scheme for the fractional diffusion-wave equation, Appl. math. model., 34, 2998-3007, (2010) · Zbl 1201.65154
[13] Hirsch, R.S., Higher order accurate difference solutions of fluid mechanics problems by a compact difference technique, J. comput. phys., 24, 90-109, (1975) · Zbl 0326.76024
[14] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16-42, (1992) · Zbl 0759.65006
[15] Cui, M.R., Compact finite difference method for the fractional diffusion equation, J. comput. phys., 228, 7792-7804, (2009) · Zbl 1179.65107
[16] Gao, G.-H.; Sun, Z.-Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. comput. phys., 230, 586-595, (2011) · Zbl 1211.65112
[17] Meerschaert, M.M.; Scheffler, H.-P.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. comput. phys., 211, 249-261, (2006) · Zbl 1085.65080
[18] 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) · Zbl 1113.65124
[19] Brunner, H.; Ling, L.; Yamamoto, M., Numerical simulations of 2D fractional subdiffusion problems, J. comput. phys., 229, 6613-6622, (2010) · Zbl 1197.65143
[20] Peaceman, D.W.; Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, J. soc. ind. appl. math., 3, 28-41, (1955) · Zbl 0067.35801
[21] Douglas, J., On the numerical integration of u_xx+uyy=ut by implicit methods, J. soc. ind. appl. math., 3, 42-65, (1955)
[22] Douglas, J.; Gunn, J.E., A general formulation of alternating direction method. part I: parabolic and hyperbolic problems, Numer. math., 6, 428-453, (1964) · Zbl 0141.33103
[23] Dendy, J.E., Alternating direction methods for nonlinear time-dependent problems, SIAM J. numer. anal., 14, 313-326, (1977) · Zbl 0365.65064
[24] Yanenko, N.N., The method of fractional steps, the solution of problems of mathematical physics in several variables, (1971), Springer-verlag · Zbl 0209.47103
[25] Ramos, J.I., Implicit, compact, linearized θ-methods with factorization for multidimensional reaction – diffusion equations, Appl. math. comput., 94, 17-43, (1998) · Zbl 0943.65098
[26] Liao, W.; Zhu, J.; Khaliq, A.Q.M., An efficient high-order algorithm for solving systems of reaction – diffusion equations, Numer. methods for partial differential eq., 18, 340-354, (2002) · Zbl 0997.65105
[27] Sun, Z.Z.; Li, X.L., A compact alternating direction implicit difference method for reaction diffusion equations, Math. numer. sinica, 27, 209-224, (2005), (in Chinese)
[28] Liao, H.-L.; Sun, Z.-Z., Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations, Numer. meth. partial differ. eq., 26, 37-60, (2010) · Zbl 1196.65154
[29] Cui, M.R., High order compact alternating direction implicit method for the generalized sine – gordon equation, J. comput. appl. math., 235, 837-849, (2010) · Zbl 1208.65126
[30] Liu, Q.X.; Liu, F.W., Modified alternating direction methds for solving a two-dimensional noncontinuous seepage flow with fractional derivatives, Math. numer. sinica, 31, 179-194, (2009), in Chinese · Zbl 1212.65342
[31] Wang, H.; Wang, K.X., An O(nlog^2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. comput. phys., 230, 7830-7839, (2011) · Zbl 1229.65165
[32] Lubich, C., Discretized fractional calculus, SIAM J. math. anal., 17, 3, 704-719, (1986) · Zbl 0624.65015
[33] Wang, H.; Wang, K.X.; Sircar, T., A direct O(nlog^2N) finite difference method for fractional diffusion equations, J. comput. phys., 229, 8095-8104, (2010) · Zbl 1198.65176
[34] Thomas, J.W., Numerical partial differential equations: finite difference methods, Texts in applied mathematics, Vol. 22, (1995), Springer-Verlag · Zbl 0831.65087
[35] Chen, C.M.; Liu, F.W.; Turner, I.; Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. comput. phys., 227, 886-897, (2007) · Zbl 1165.65053
[36] Sewell, G., The numerical solution of ordinary and partial differential equations, (2005), John Wiley & Sons · Zbl 1089.65053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.