×

zbMATH — the first resource for mathematics

A second order explicit finite difference method for the fractional advection diffusion equation. (English) Zbl 1268.65118
Summary: We develop a numerical method for fractional advection diffusion problems with source terms in domains with homogeneous boundary conditions. The numerical method is derived by using a Lax-Wendroff-type time discretization procedure, it is explicit and second order accurate. The convergence of the numerical method is studied and numerical results are presented.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
45K05 Integro-partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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
[2] Metzler, R.; Klafter, J., Accelerating Brownian motion: a fractional dynamics approach to fast diffusion, Europhys. lett., 51, 492-498, (2000)
[3] Chaves, A.S., A fractional diffusion equation to describe Lévy flights, Phys. lett. A, 239, 13-16, (1998) · Zbl 1026.82524
[4] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., Application of a fractional advection – dispersion equation, Water resour. res., 36, 1403-1412, (2000)
[5] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., The fractional order governing equation of Lévy motion, Water resour. res., 36, 1413-1423, (2000)
[6] 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
[7] Sousa, E., Finite difference approximations for a fractional advection diffusion problem, J. comput. phys., 228, 4038-4054, (2009) · Zbl 1169.65126
[8] 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
[9] Zhang, X.; Mouchao, L.; Crawford, J.W.; Young, I.M., The impact of boundary on the fractional advection dispersion equation for solute transport in soil: defining the fractional dispersive flux with the Caputo derivatives, Adv. water resour., 30, 1205-1217, (2007)
[10] 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
[11] Deng, W., Finite element method for the space and time fractional fokker – planck equation, SIAM J. numer. anal., 47, 204-226, (2008) · Zbl 1416.65344
[12] Ervin, V.J.; Norbert, H.; Roop, J.P., Numerical approximation of a time dependent nonlinear, space-fractional diffusion equation, SIAM J. numer. anal., 45, 572-591, (2007) · Zbl 1141.65089
[13] Huang, Q.; Huang, G.; Zhan, H., A finite element solution for the fractional advection – dispersion equation, Adv. water resour., 31, 1578-1589, (2008)
[14] Li, C.; Zhao, Z.; Chen, Y., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. math. appl., 62, 855-875, (2011) · Zbl 1228.65190
[15] Roop, J.P., Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \(\mathbb{R}^2\), J. comput. appl. math., 193, 243-268, (2006) · Zbl 1092.65122
[16] Zhang, H.; Fawang, L.; Anh, V., Galerkin finite element approximation of symmetric space-fractional partial differential equations, Appl. math. comput., 217, 2534-2545, (2010) · Zbl 1206.65234
[17] Zheng, Y.Y.; Li, C.P.; Zhao, Z.G., A fully discrete discontinuous Galerkin method for nonlinear fractional fokker – planck equation, Math. probl. eng., (2010), Article ID 279038
[18] Lax, P.D.; Wendroff, B., Difference schemes for hyperbolic equations with high order of accuracy, Commun. pur. appl. math., 17, 381-398, (1964) · Zbl 0233.65050
[19] Diethelm, K., The analysis of fractional differential equations, Lecture notes in math., (2004) · Zbl 1085.41020
[20] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier · Zbl 1092.45003
[21] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Dover · Zbl 0428.26004
[22] Ortigueira, M.D., Fractional calculus for scientists and engineers, () · Zbl 1251.26005
[23] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[24] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Science Publishers · Zbl 0818.26003
[25] Sousa, E., Numerical approximations for fractional diffusion equations via splines, Comput. math. appl., 62, 938-944, (2011) · Zbl 1228.65153
[26] E. Sousa, C. Li, A weighted finite difference method for the fractional diffusion based on the Riemann-Liouville derivative, 2011. arXiv:1109.2345v1 [math.NA] (submitted for publication). · Zbl 1326.65111
[27] Sousa, E., On the edge of stability analysis, Appl. numer. math., 59, 1322-1336, (2009) · Zbl 1166.65045
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.