Numerical methods and analysis for a class of fractional advection-dispersion models. (English) Zbl 1268.65124

Summary: A class of fractional advection-dispersion models (FADMs) is considered. These models include five fractional advection-dispersion models, i.e., the time FADM, the mobile/immobile time FADM with a time Caputo fractional derivative \(0<\gamma<1\), the space FADM with two sides Riemann-Liouville derivatives, the time-space FADM and the time fractional advection-diffusion-wave model with damping with index \(1<\gamma<2\). These equations can be used to simulate the regional-scale anomalous dispersion with heavy tails. We propose computationally effective implicit numerical methods for these FADMs. The stability and convergence of the implicit numerical methods are analysed and compared systematically. Finally, some results are given to demonstrate the effectiveness of theoretical analysis.


65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
45K05 Integro-partial differential equations


Full Text: DOI


[1] Risken, H., The Fokker-Planck Equation (1988), Springer: Springer Berlin
[2] Zhang, Y.; Benson, D. A.; Reeves, D. M., Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications, Advances in Water Resources, 32, 561-581 (2009)
[3] Adams, E. E.; Gelhar, L. W., Field study of dispersion in a heterogeneous aquifer: 2. Spatial moment analysis, Water Resources Research, 28, 12, 3293-3307 (1992)
[4] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Application of a fractional advection-dispersion equation, Water Resources Research, 36, 6, 1403-1412 (2000)
[5] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., The fractional-order governing equation of Levy motion, Water Resources Research, 36, 6, 1413-1423 (2000)
[6] Eggleston, J.; Rojstaczer, S., Identification of large-scale hydraulic conductivity trends and the influence of trends on contaminant transport, Water Resources Researces, 34, 9, 2155-2168 (1998)
[7] Major, E.; Benson, D. A.; Revielle, J.; Ibrahim, H.; Dean, A. M.; Maxwell, R. M.; Poeter, E. P.; Dogan, M., Comparison of Fickian and temporally non-local transport theories over many scales in an exhaustively sampled sandstone slab, Water Resource Research, 47, W10519 (2011)
[8] Bouchaud, J. P.; Georges, A., Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Physics Reports (Review Section of Physics Letters), 195, 4-5, 127-293 (1990)
[9] Mainardi, F., Fraction calculus: some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractal and Fractional Calin Continuum Mechanics (1997), Springer: Springer Wien), 291-348 · Zbl 0917.73004
[10] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractielonal Fokker-Planck equation, Journal of Computational and Applied Mathematics, 166, 209-219 (2004) · Zbl 1036.82019
[11] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Applied Mathematics and Computation, 91, 12-20 (2007) · Zbl 1193.76093
[12] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Applied Numerical Mathematics, 56, 80-90 (2006) · Zbl 1086.65087
[13] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis, 4, 153-1925 (2001) · Zbl 1054.35156
[14] Podlubny, I., Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis, 3, 4, 359-386 (2000) · Zbl 1030.26011
[15] Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y. Q.; Vinagre Jara, B. M., Matrix approach to discrete fractional calculus II: partial fractional differential equations, Journal of Computational Physics, 228, 8, 3137-3153 (2009), http://dx.doi.org/10.1016/j.jcp.2009.01.014 · Zbl 1160.65308
[16] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[17] Meerschaert, M. M.; Scheffler, H. P., Limit theorems for continuous time random walks with infinite mean waiting times, Journal of Applied Probability, 41, 3, 623-638 (2004) · Zbl 1065.60042
[18] Becker-Kern, P.; Meerschaert, M. M.; Scheffler, H. P., Limit theorem for continuous time random walks with two time scales, Journal of Applied Probability, 41, 455-466 (2004) · Zbl 1050.60038
[19] Schumer, R.; Benson, D. A.; Meerschaert, M. M.; Baeumer, B., Fractal mobile/immobile solute transport, Water Resources Researces, 39, 10, 1296 (2003)
[20] Zhang, Y.; Benson, D. A.; Meerschaert, M. M.; Scheffler, H. P., On using random walks to solve the space-fractional advection-dispersion equations, Journal of Statistical Physics, 123, 1, 89 (2006) · Zbl 1092.82038
[21] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[22] Yang, Q.; Liu, F.; Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathematical Modelling, 34, 1, 200-218 (2010) · Zbl 1185.65200
[23] Lin, R.; Liu, F., Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Analysis, 66, 856-869 (2007) · Zbl 1118.65079
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