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Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. (English) Zbl 1191.65116
This paper is concerned with approximation of solutions for anomalous sub-diffusion equations in planar domains. Two numerical methods (one explicit and another implicit) are proposed that rely on the finite difference method. The paper focuses on the stability and convergence of the method. Further, a new multivariate extrapolation is introduced to improve the accuracy of the methods. Various numerical examples are presented to support the theoretical results obtained in this paper.
MSC:
65M12Stability and convergence of numerical methods (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
35K05Heat equation
References:
[1]Compte, A.: Continuous time random walks on moving fluids. Phys. Rew. E 55, 6821–6831 (1997) · doi:10.1103/PhysRevE.55.6821
[2]Compte, A., Caceres, M.O.: Fractional dymamics in random velocity fields. Phys. Rev. Lett. 81, 3140–3143 (1998) · doi:10.1103/PhysRevLett.81.3140
[3]Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Physica A 276, 448–455 (2000) · doi:10.1016/S0378-4371(99)00469-0
[4]Baeumer, B., Meerschaert, M.M., Benson, D.A., Wheatsraft, S.W.: Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37, 1543–1550 (2001) · doi:10.1029/2000WR900409
[5]Chen, C.-M., Liu, F., Turner, I., Anh, V.: A new Fourier analysis method for the Galilei invariant fractional advection diffusion equation. ANZIAM J. 48(CTAC2006), C775–C789 (2007)
[6]Chen, C.-M., Liu, F., Turner, I., Anh, V.: Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886–897 (2007) · Zbl 1165.65053 · doi:10.1016/j.jcp.2007.05.012
[7]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 · doi:10.1016/j.amc.2007.09.020
[8]Chen, C.-M., Liu, F., Anh, V.: Numerical analysis of the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives. Appl. Math. Comput. 204, 340–351 (2008) · Zbl 1153.76010 · doi:10.1016/j.amc.2008.06.052
[9]Chen, C.-M., Liu, F., Anh, V.: A Fourier method and an extrapolation technique for Stokes first problem for a heated generalized second grade fluid with fractional derivative. J. Comput. Appl. Math. 223, 777–789 (2009) · Zbl 1153.76049 · doi:10.1016/j.cam.2008.03.001
[10]Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673–696 (2006) · Zbl 1090.65147 · doi:10.1090/S0025-5718-06-01788-1
[11]Enson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-despersion equation. Water Resour. Res. 36(6), 1403–1412 (2000) · doi:10.1029/2000WR900031
[12]Ervin, V.S., Roop, J.P., Paul, J.: Variational solution of fractional advection dispersion equations on bounded domains in R-d. Numer. Meth. Part. D. E. 23, 256–281 (2007) · Zbl 1117.65169 · doi:10.1002/num.20169
[13]Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)
[14]Liu, F., Shen, S., Anh, V., Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. ANZIAM J. 46(E), 488–504 (2005)
[15]Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12–20 (2007) · Zbl 1193.76093 · doi:10.1016/j.amc.2006.08.162
[16]Cushman, J.H., Cinn, T.R.: Fractional advection-disperision equation: a classical mass balance with convolution-Fickian flux. Water Resour. Res. 36, 3763–3766 (2000) · doi:10.1029/2000WR900261
[17]Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2. J. Comput. Appl. Math. 193, 243–268 (2006) · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005
[18]Seki, K., Wojcik, M., Tachiya, M.: Fractional reaction-diffusion equation. J. Chem. Phys. 119, 2165–2174 (2003) · doi:10.1063/1.1587126
[19]Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004). · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[20]Meerschaert, M.M., Nane, E., Xiao, Y.: Large deviations for local time fractional Brownian motion and applications. J. Math. Anal. Appl. 346, 432–445 (2008) · Zbl 1147.60025 · doi:10.1016/j.jmaa.2008.05.087
[21]Giona, M., Roman, H.E.: Fractional diffusion equation for transport phenomena in randam media. Physica A 185, 87–97 (1992) · doi:10.1016/0378-4371(92)90441-R
[22]Zhuang, P., Liu, F.: Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22(3), 87–99 (2006) · Zbl 1140.65094 · doi:10.1007/BF02832039
[23]Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation. SIAM J. Numer. Anal. 46(2), 1079–1095 (2008) · Zbl 1173.26006 · doi:10.1137/060673114
[24]Metzler, R., Klafter, J., Sokolov, I.M.: Anomalous transport in external fields: continuous time random walksand fractional diffusion equations extended. Phys. Rev. E 58, 1621–1633 (1998) · doi:10.1103/PhysRevE.58.1621
[25]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) · doi:10.1103/PhysRevLett.82.3563
[26]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 · doi:10.1016/S0370-1573(00)00070-3
[27]Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Physica A 278, 107–125 (2000) · doi:10.1016/S0378-4371(99)00503-8
[28]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) · doi:10.1016/S0301-0104(02)00714-0
[29]Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von neumman-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862–1874 (2005) · Zbl 1119.65379 · doi:10.1137/030602666
[30]Shen, S., Liu, F.: Error analysis of an explicit finite difference approximation for the space fractional diffusion. ANZIAM J. 46(E), 871–887 (2005)
[31]Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comp. Phys. 205, 719–736 (2005) · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025
[32]Balakrishnan, V.: Anomalous diffusion in one dimension. Physica A 132, 569–580 (1985) · Zbl 0654.60065 · doi:10.1016/0378-4371(85)90028-7
[33]McLean, W., Mustapha, K.: Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algor. (2009). doi: 10.1007/s11075-008-9258-8
[34]Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989) · Zbl 0692.45004 · doi:10.1063/1.528578
[35]Wyss, W.: Fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986) · Zbl 0632.35031 · doi:10.1063/1.527251
[36]Zhang, Y., Meerschaert, M.M., Baeumer, B.: Particle tracking for time-fractional diffusion. Phys. Rev. E 78, 036705 (2008) · doi:10.1103/PhysRevE.78.036705