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Finite difference approximations for a fractional advection diffusion problem. (English) Zbl 1169.65126

Author’s abstract: The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion.

We consider a one-dimensional advection-diffusion model, where the usual second-order derivative gives place to a fractional derivative of order α, with 1<α2. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.

MSC:
65R20Integral equations (numerical methods)
45K05Integro-partial differential equations
26A33Fractional derivatives and integrals (real functions)
References:
[1]Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M.: Application of a fractional advection – dispersion equation, Water resour. Res. 36, 1403-1412 (2000)
[2]Huang, G.; Huang, Q.; Zhan, H.: Evidence of one-dimensional scale-dependent fractional advection – dispersion, J. contam. Hydrol. 85, 53-71 (2006)
[3]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
[4]Metzler, R.; Klafter, J.: Accelerating Brownian motion: a fractional dynamics approach to fast diffusion, Europhys. lett. 51, 492-498 (2000)
[5]Pachepsky, Y.; Benson, D.; Rawls, W.: Simulating scale-dependent solute transport in soils with the fractional advective – dispersive equation, Soil sci. Soc. am. J. 4, 1234-1243 (2000)
[6]Zhou, L.; Selim, H. M.: Application of the fractional advection – dispersion equation in porous media, Soil sci. Soc. am. J. 67, 1079-1084 (2003)
[7]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) · Zbl 1165.65053 · doi:10.1016/j.jcp.2007.05.012
[8]Shen, S.; Liu, F.: Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends, Anziam j. 46, No. E, C871-C887 (2005) · Zbl 1078.65563 · doi:http://anziamj.austms.org.au/V46/CTAC2004/Shen/
[9]Tadjeran, C.; Meerschaert, M. M.; Scheffler, H. -P.: A second-order accurate numerical approximation for the fractional diffusion equation, J. comput. Phys. 213, 205-213 (2006) · Zbl 1089.65089 · doi:10.1016/j.jcp.2005.08.008
[10]Tadjeran, C.; Meerschaert, M. M.: A second-order accurate numerical approximation for the two-dimensional fractional diffusion equation, J. comput. Phys. 220, 813-823 (2007) · Zbl 1113.65124 · doi:10.1016/j.jcp.2006.05.030
[11]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 · doi:10.1137/030602666
[12]Yuste, S. B.: Weighted average finite difference methods for fractional diffusion equations, J. comput. Phys. 216, 264-274 (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006
[13]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, Appl. math. Comput. 191, 12-20 (2006) · Zbl 1193.76093 · doi:10.1016/j.amc.2006.08.162
[14]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
[15]Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M.: The fractional order governing equation of Lévy motion, Water resour. Res. 36, 1413-1423 (2000)
[16]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[17]Podlubny, I.: Fractional differential equations, (1999)
[18]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[19]Lax, P. D.; Wendroff, B.: Difference schemes for hyperbolic equations with high order of accuracy, Commun. pure appl. Math. 17, 381-398 (1964) · Zbl 0233.65050 · doi:10.1002/cpa.3160170311
[20]Sousa, E.: The controversial stability analysis, Appl. math. Comput. 145, 777-794 (2003) · Zbl 1032.65103 · doi:10.1016/S0096-3003(03)00274-1
[21]Morton, K. W.: Numerical solution of convection diffusion problems, (1996)
[22]Warming, F. F.; Hyett, B. J.: The modified equation approach to the stability and accuracy analysis of finite difference methods, J. comput. Phys. 14, 159-179 (1974) · Zbl 0291.65023 · doi:10.1016/0021-9991(74)90011-4
[23]Sod, G. A.: Numerical methods in fluid dynamics: initial and initial boundary-value problems, (1988)
[24]Mcculloch, J. H.; Panton, D. P.: Precise tabulation of the maximally-skewed stable distributions and densities, Comput. stat. Data anal. 23, 307-320 (1997) · Zbl 0875.62631 · doi:10.1016/S0167-9473(96)00039-4
[25]A. Ogata, R.B. Banks, A solution of the differential equation of longitudinal dispersion in porous media, US Geol. Surv., Prof. Paper No. 411-A, 1961, pp. A1 – A7.
[26]Hirsch, C.: Numerical computations of internal and external flows, Numerical computations of internal and external flows (1990)
[27]Smith, G. D.: Numerical solution of partial differential equations, (1985)