Finite difference approximations for fractional advection-dispersion flow equations

Journal of Computational and Applied Mathematics

Volumn 172, Issue 1, 2004, Pages 65-77

  Meerschaert, Mark M ^a   Tadjeran, Charles ^a  

^a UNIVERSITY OF NEVADA   (United States)

Author keywords

Backward Euler method; Finite difference approximation; Fractional advection dispersion; Fractional derivative; Fractional diffusion; Implicit Euler method; Numerical fractional ADE; Radial advection; Radial dispersion; Stability

Indexed keywords

APPROXIMATION THEORY; DISPERSIONS; FINITE DIFFERENCE METHOD; FLOW OF FLUIDS; POROSITY;

ADVECTION; FINITE DIFFERENCE APPROXIMATIONS;

COMPUTATIONAL METHODS;

ADVECTION-DIFFUSION EQUATION; FINITE DIFFERENCE TECHNIQUE; FLUID FLOW; GROUNDWATER; POLLUTANT TRANSPORT; POROUS MEDIUM;

EID: 4444368867     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2004.01.033     Document Type: Article

References (35)

1. Baeumer B. Meerschaert M.M. Stochastic solutions for fractional Cauchy problems Frac. Calc. Appl. Anal. 4 2001 481-500
2. Baeumer B. Meerschaert M.M. Benson D.A. Wheatcraft S.W. Subordinated advection-dispersion equation for contaminant transport Water Resour. Res. 37 2001 1543-1550
3. Barkai E. Metzler R. Klafter J. From continuous time random walks to the fractional Fokker-Planck equation Phys. Rev. E 61 2000 132-138
4. Benson D. Schumer R. Meerschaert M. Wheatcraft S. Fractional dispersion, Lévy motions, and the MADE tracer tests Transport Porous Media 42 2001 211-240
5. Benson D. Wheatcraft S. Meerschaert M. Application of a fractional advection-dispersion equation Water Resour. Res. 36 2000 1403-1412
6. Benson D. Wheatcraft S. Meerschaert M. The fractional-order governing equation of Lévy motion Water Resour. Res. 36 2000 1413-1424
7. Blumen A. Zumofen G. Klafter J. Transport aspects in anomalous diffusion: Lévy walks Phys. Rev. A 40 1989 3964-3973
8. Bouchaud J.P. Georges A. Anomalous diffusion in disordered media-statistical mechanisms, models and physical applications Phys. Rep. 195 1990 127-293
9. R. Carroll, et al., Project Shoal areas tracer experiment, Desert Research institute, Division of Hydrologic Sciences Report No. 45177, 2000.
10. Chaves A. Fractional diffusion equation to describe Lévy flights Phys. Lett. A 239 1998 13-16
11. Chechkin A.V. Klafter J. Sokolov I.M. Fractional Fokker-Planck equation for ultraslow kinetics Europhys. Lett. 63 2003 326-332
12. Cushman J.H. Ginn T.R. Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux Water Resour. Res. 36 2000 3763-3766
13. G.J. Fix, J.P. Roop, Least squares finite element solution of a fractional order two-point boundary value problem, Comput. Math. Appl. 2003, to appear.
14. R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional calculus and continuous-time finance. III, The diffusion limit. Mathematical finance (Konstanz, 2000), Trends in Math., Birkhuser, Basel, 2001, pp. 171-180.
15. Herrick M. Benson D. Meerschaert M. McCall K. Hydraulic conductivity, velocity, and the order of the fractional dispersion derivative in a highly heterogeneous system Water Resour. Res. 38 2002 1227-1239
16. Klafter J. Blumen A. Shlesinger M.F. Stochastic pathways to anomalous diffusion Phys. Rev. A 35 1987 3081-3085
17. F. Liu, V. Ahn, I. Turner, Numerical Solution of the Fractional Advection-Dispersion Equation, 2002, preprint.
18. Meerschaert M. Benson D. Baeumer B. Operator Lévy motion and multiscaling anomalous diffusion Phys. Rev. E 63 2001 1112-1117
19. Meerschaert M. Benson D. Scheffler H.P. Baeumer B. Stochastic solution of space-time fractional diffusion equations Phys. Rev. E 65 2002 1103-1106
20. Meerschaert M. Benson D. Scheffler H.P. Becker-Kern P. Governing equations and solutions of anomalous random walk limits Phys. Rev. E 66 2002 102-105
21. Meerschaert M.M. Scheffler H.P. Semistable Lévy Motion Frac. Calc. Appl. Anal. 5 2002 27-54
22. Miller K. Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 Wiley New York
23. Podlubny I. Fractional Differential Equations 1999 Academic Press New York
24. Raberto M. Scalas E. Mainardi F. Waiting-times and returns in high-frequency financial data: an empirical study Physica A 314 2002 749-755
25. Reimus P. Pohll G. Mihevc T. Chapman J. Haga M. Lyles B. Kosinski S. Niswonger R. Sanders P. Testing and parameterizing a conceptual model for solute transport in a fractured granite using multiple tracers in a forced-gradient test Water Resour. Res. 39 2003 1356-1370
26. Richtmyer R.D. Morton K.W. Difference Methods for Initial-Value Problems 1994 Krieger Publishing Malabar, Florida
27. Sabatelli L. Keating S. Dudley J. Richmond P. Waiting time distributions in financial markets Eur. Phys. J. B 27 2002 273-275
28. Saichev A.I. Zaslavsky G.M. Fractional kinetic equations: solutions and applications Chaos 7 1997 753-764
29. Samko S. Kilbas A. Marichev O. Fractional Integrals and Derivatives: Theory and Applications 1993 Gordon and Breach London
30. Scalas E. Gorenflo R. Mainardi F. Fractional calculus and continuous-time finance Phys. A 284 2000 376-384
31. Schumer R. Benson D.A. Meerschaert M.M. Wheatcraft S.W. Eulerian derivation of the fractional advection-dispersion equation J. Contaminant Hydrol. 48 2001 69-88
32. Schumer R. Benson D.A. Meerschaert M.M. Baeumer B. Multiscaling fractional advection-dispersion equations and their solutions Water Resour. Res. 39 2003 1022-1032
33. C. Tadjeran, D.A. Benson, M.M. Meerschaert, Fractional Radial flow and Its Application to Field Data, 2003, available at http://unr.edu/homepage/mcubed/frade_wrr.pdf
34. Tuan V.K. Gorenflo R. Extrapolation to the limit for numerical fractional differentiation Z. agnew. Math. Mech. 75 1995 646-648
35. Zaslavsky G. Fractional kinetic equation for Hamiltonian chaos. Chaotic advection, tracer dynamics and turbulent dispersion Phys. D 76 1994 110-122