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The high-order compact numerical algorithms for the two-dimensional fractional sub-diffusion equation. (English) Zbl 1410.65315

Summary: In this paper, performing the average operators on the space variables, a numerical scheme with third-order temporal convergence for the two-dimensional fractional sub-diffusion equation is considered, for which the unconditional stability and convergence in \(L_{1}(L_{\infty})\)-norm are strictly analyzed for \(\alpha\in (0, 0.9569347]\) by using the discrete energy method. Therewith, adding small perturbation terms, we construct a compact alternating direction implicit difference scheme for the two-dimensional case. Finally, some numerical results have been given to show the computational efficiency and numerical accuracy of both schemes for all \(\alpha\in (0, 1)\).

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
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[1] Ervin, V. J.; Roop, J. P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differ. Equ., 22, 558-576 (2006) · Zbl 1095.65118
[2] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Finite element approximation for a modified anomalous subdiffusion equation, Appl. Math. Model., 35, 4103-4116 (2011) · Zbl 1221.65257
[3] Zeng, F.; Li, C.; Liu, F.; Turner, I., Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM J. Sci. Comput., 37, A55-A78 (2015) · Zbl 1334.65162
[4] Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121
[5] Li, X.; Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47, 2108-2131 (2009) · Zbl 1193.35243
[6] Zheng, M.; Liu, F.; Turner, I.; Anh, V., A novel high order space-time spectral method for the time-fractional Fokker-Planck equation, SIAM J. Sci. Comput., 37, A701-A724 (2015) · Zbl 1320.82052
[7] 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) · Zbl 1231.65178
[8] Liu, Q.; Gu, Y.; Zhuang, P.; Liu, F.; Nie, Y., An implicit RBF meshless approach for time fractional diffusion equations, Comput. Mech., 48, 1-12 (2011) · Zbl 1377.76025
[9] Bechelova, A. R., On the convergence of difference schemes for the diffusion equation of fractional order, Ukrainian Math. J., 50, 7, 1131-1134 (1998) · Zbl 0934.65100
[10] 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
[11] 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
[12] Sun, Z. Z.; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 193-209 (2006) · Zbl 1094.65083
[13] 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) · Zbl 1191.65116
[14] Zhang, Y.; Sun, Z. Z., Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230, 8713-8728 (2011) · Zbl 1242.65174
[15] Cui, M., Combined compact difference scheme for the time fractional convection-diffusion equation with variable coefficients, Appl. Math. Comput., 246, 464-473 (2014) · Zbl 1339.65106
[16] Chen, J.; Liu, F.; Liu, Q.; Chen, X.; Anh, V.; Turner, I.; Burrage, K., Numerical simulation for the three-dimension fractional sub-diffusion equation, Appl. Math. Model., 38, 3695-3705 (2014) · Zbl 1429.65179
[17] 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
[18] Yuste, S. B., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216, 264-274 (2006) · Zbl 1094.65085
[19] 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) · Zbl 1119.65379
[20] Mohebbi, A.; Abbaszadeha, 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) · Zbl 1287.65064
[21] Gao, G.; Sun, Z. Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230, 586-595 (2011) · Zbl 1211.65112
[22] Ren, J.; Sun, Z. Z.; Zhao, X., Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 232, 456-467 (2013) · Zbl 1291.35428
[23] Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228, 7792-7804 (2009) · Zbl 1179.65107
[24] 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) · Zbl 1173.26006
[25] Gao, G.; Sun, Z. Z.; Zhang, H. W., A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259, 33-50 (2014) · Zbl 1349.65088
[26] 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) · Zbl 1349.65092
[27] Alikhanov, A. A., A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280, 424-438 (2015) · Zbl 1349.65261
[28] Wang, Z.; Vong, S., Compact difference schemes for the modified anomalous fractional subdiffusion equation and the fractional diffusion-wave equation, J. Comput. Phys., 277, 1-15 (2014) · Zbl 1349.65348
[29] Li, C. P.; Ding, H. F., Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model., 38, 3802-3821 (2014) · Zbl 1429.65188
[31] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[32] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[33] Kilbas, A.; Srivastava, H.; Trujillo, J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Boston · Zbl 1092.45003
[34] 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
[35] Tian, W.; Zhou, H.; Deng, W. H., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp., 84, 1703-1727 (2015) · Zbl 1318.65058
[36] 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) · Zbl 1196.65154
[37] Gao, G.; Sun, Z. Z., Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations, Comput. Math. Appl., 69, 926-948 (2015) · Zbl 1443.65124
[39] Laub, A. J., Matrix Analysis for Scientists and Engineers, Society for Industrial and Applied Mathematics (SIAM) (2005), Philadelphia, PA · Zbl 1077.15001
[40] Sun, Z. Z., Numerical Methods of Partial Differential Equations (2012), Science Press: Science Press Beijing
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