×
Meerschaert, Mark M.; Scheffler, Hans-Peter; Tadjeran, Charles

Finite difference methods for two-dimensional fractional dispersion equation. (English) Zbl 1085.65080

J. Comput. Phys. 211, No. 1, 249-261 (2006).
Summary: Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of application. The authors discuss a practical alternating directions implicit method to solve a class of two-dimensional initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. First-order consistency, unconditional stability, and (therefore) first-order convergence of the method are proven using a novel shifted version of the classical Grünwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.

Cited in 1 Review
Cited in 206 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
35G25 Initial value problems for nonlinear higher-order PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35G30 Boundary value problems for nonlinear higher-order PDEs

Keywords:

Two-dimensional fractional partial differential equation; Implicit Euler method; Multi-dimensional fractional PDE; Numerical fractional PDE; Alternating direction implicit methods; initial-boundary value problem; consistency; stability; convergence; Grünwald finite difference approximation; numerical example
PDF BibTeX XML Cite
\textit{M. M. Meerschaert} et al., J. Comput. Phys. 211, No. 1, 249--261 (2006; Zbl 1085.65080)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Baeumer, B.; Meerschaert, M.M., Stochastic solutions for fractional Cauchy problems, Frac. calc. appl. anal., 4, 481-500, (2001) · Zbl 1057.35102
[2] Baeumer, B.; Meerschaert, M.M.; Benson, D.A.; Wheatcraft, S.W., Subordinated advection-dispersion equation for contaminant transport, Water resour. res., 37, 1543-1550, (2001)
[3] Baeumer, B.; Benson, D.A.; Meerschaert, M.M., Advection and dispersion in time and space, Physica A, 350, 245-262, (2005)
[4] Barkai, E.; Metzler, R.; Klafter, J., From continuous time random walks to the fractional Fokker-Planck equation, Phys. rev. E, 61, 132-138, (2000)
[5] Benson, D.A.; Wheatcraft, S.; Meerschaert, M.M., Application of a fractional advection-dispersion equation, Water resour. res., 36, 1403-1412, (2000)
[6] Benson, D.A.; Schumer, R.; Meerschaert, M.M.; Wheatcraft, S.W., Fractional dispersion, Lévy motions, and the MADE tracer tests, Transp. porous media, 42, 211-240, (2001)
[7] Benson, D.A.; Tadjeran, C.; Meerschaert, M.M.; Farnham, I.; Pohll, G., Radial fractional-order dispersion through fractured rock, Water resour. res., 40, 1-9, (2004)
[8] Chechkin, A.V.; Klafter, J.; Sokolov, I.M., Fractional Fokker-Planck equation for ultraslow kinetics, Europhys. lett., 63, 326-332, (2003)
[9] Cushman, J.H.; Ginn, T.R., Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux, Water resour. res., 36, 3763-3766, (2000)
[10] Deng, Z.; Singh, V.P.; Bengtsson, L., Numerical solution of fractional advection-dispersion equation, J. hydraul. eng., 130, 422-431, (2004)
[11] J.S. Duan, Time- and space-fractional partial differential equations. J. Math. Phys. 46 (2005) 013504 (8 pp.). doi:10.1063/1.1819524. · Zbl 1076.26006
[12] Douglas, J.; Kim, S., Improved accuracy for locally one-dimensional methods for parabolic equations, Math. models meth. appl. sci., 11, 9, 1563-1579, (2001) · Zbl 1012.65095
[13] V.J. Ervin, J.P. Roop, Variational solution to the fractional advection dispersion equation, Numer. Meth. P.D.E., to appear, (2005).
[14] V.J. Ervin, J.P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in Rd (2005), preprint.
[15] Fix, G.J.; Roop, J.P., Least squares finite element solution of a fractional order two-point boundary value problem, Comput. math. appl., 48, 1017-1033, (2004) · Zbl 1069.65094
[16] Gorenflo, R.; Mainardi, F.; Scalas, E.; Raberto, M., Fractional calculus and continuous-time finance. III. the diffusion limit. mathematical finance (Konstanz, 2000), Trends math., 171-180, (2001) · Zbl 1138.91444
[17] Isaacson, E.; Keller, H.B., Analysis of numerical methods, (1966), Wiley New York · Zbl 0168.13101
[18] 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
[19] Liu, F.; Ahn, V.; Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. comput. appl. math., 166, 209-219, (2004) · Zbl 1036.82019
[20] Lynch, V.E.; Carreras, B.A.; del-Castillo-Negrete, D.; Ferreira-Mejias, K.M.; Hicks, H.R., Numerical methods for the solution of partial differential equations of fractional order, J. comput. phys., 192, 406-421, (2003) · Zbl 1047.76075
[21] Mainardi, F.; Raberto, M.; Gorenflo, R.; Scalas, E., Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A, 287, 468-481, (2000)
[22] Meerschaert, M.M.; Benson, D.A.; Scheffler, H.P.; Baeumer, B., Stochastic solution of space-time fractional diffusion equations, Phys. rev. E, 65, 1103-1106, (2002)
[23] Meerschaert, M.M.; Scheffler, H.P., Semistable Lévy motion, Fract. calc. appl. anal., 5, 27-54, (2002) · Zbl 1032.60043
[24] Meerschaert, M.M.; Benson, D.A.; Scheffler, H.P.; Becker-Kern, P., Governing equations and solutions of anomalous random walk limits, Phys. rev. E, 66, 102-105, (2002)
[25] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. comput. appl. math., 172, 65-77, (2003) · Zbl 1126.76346
[26] Meerschaert, M.M.; Mortensen, J.; Scheffler, H.P., Vector Grünwald formula for fractional derivatives, Fract. calc. appl. anal., 7, 61-81, (2004) · Zbl 1084.65024
[27] M.M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., on-line articles in press. · Zbl 1086.65087
[28] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. phys. A, 37, R161-R208, (2004) · Zbl 1075.82018
[29] Miller, K.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[30] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[31] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[32] Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A, 314, 749-755, (2002) · Zbl 1001.91033
[33] Richtmyer, R.D.; Morton, K.W., Difference methods for initial-value problems, (1994), Krieger Malabar, FL · Zbl 0155.47502
[34] Sabatelli, L.; Keating, S.; Dudley, J.; Richmond, P., Waiting time distributions in financial markets, Eur. phys. J. B, 27, 273-275, (2002)
[35] Scalas, E.; Gorenflo, R.; Mainardi, F., Fractional calculus and continuous-time finance, Physica A, 284, 376-384, (2000)
[36] Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Wheatcraft, S.W., Eulerian derivation of the fractional advection-dispersion equation, J. contamin. hydrol., 48, 69-88, (2001)
[37] Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Baeumer, B., Multiscaling fractional advection-dispersion equations and their solutions, Water resour. res., 39, 1022-1032, (2003)
[38] Tuan, V.K.; Gorenflo, R., Extrapolation to the limit for numerical fractional differentiation, Z. angew. math. mech., 75, 646-648, (1995) · Zbl 0860.65011
[39] Samko, S.; Kilbas, A.; Marichev, O., Fractional integrals and derivatives: theory and applications, (1993), Gordon & Breach London · Zbl 0818.26003
[40] 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, 5, 1862-1874, (2005) · Zbl 1119.65379
[41] Varga, R., Matrix iterative analysis, (1962), Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.