Finite difference approximations for a fractional advection diffusion problem

Journal of Computational Physics

Volumn 228, Issue 11, 2009, Pages 4038-4054

  Sousa, Ercília ^a  

^a UNIVERSITY OF COIMBRA   (Portugal)

Author keywords

Finite differences; Fractional advection diffusion; Stability

Indexed keywords

BROWNIAN MOVEMENT; DIFFUSION; DISPERSIONS; FINITE DIFFERENCE METHOD; PARTIAL DIFFERENTIAL EQUATIONS;

A-PARTICLES; ADVECTION-DIFFUSION EQUATION; ADVECTION-DIFFUSION PROBLEM; ANOMALOUS DIFFUSION; ANOMALOUS DISPERSION; DIFFUSION MODEL; FINITE DIFFERENCE; FINITE DIFFERENCE APPROXIMATIONS; FRACTIONAL ADVECTION-DIFFUSIONS; FRACTIONAL DERIVATIVES;

ADVECTION;

EID: 64049113904     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.02.011     Document Type: Article

References (27)

1. Benson D.A., Wheatcraft S.W., and Meerschaert M.M. Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (2000) 1403-1412
2. Huang G., Huang Q., and Zhan H. Evidence of one-dimensional scale-dependent fractional advection-dispersion. J. Contam. Hydrol. 85 (2006) 53-71
3. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1-77
4. Metzler R., and Klafter J. Accelerating Brownian motion: a fractional dynamics approach to fast diffusion. Europhys. Lett. 51 (2000) 492-498
5. Pachepsky Y., Benson D., and Rawls W. Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation. Soil Sci. Soc. Am. J. 4 (2000) 1234-1243
6. Zhou L., and Selim H.M. Application of the fractional advection-dispersion equation in porous media. Soil Sci. Soc. Am. J. 67 (2003) 1079-1084
7. Chen C.-M., Liu F., Turner I., and Anh V. A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227 (2007) 886-897
8. Shen S., and Liu F. Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends. ANZIAM J. 46 E (2005) C871-C887
9. Tadjeran C., Meerschaert M.M., and Scheffler H.-P. A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213 (2006) 205-213
10. Tadjeran C., and Meerschaert M.M. A second-order accurate numerical approximation for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007) 813-823
11. Yuste S.B., and Acedo L. An explicit finite difference method and a new von Neumann type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42 (2005) 1862-1874
12. Yuste S.B. Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216 (2006) 264-274
13. Liu F., Zhuang P., Anh V., Turner I., and Burrage K. Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191 (2006) 12-20
14. Meerschaert M.M., and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172 (2004) 65-77
15. Benson D.A., Wheatcraft S.W., and Meerschaert M.M. The fractional order governing equation of Lévy motion. Water Resour. Res. 36 (2000) 1413-1423
16. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and Applications of Fractional Differential equations (2006), Elsevier
17. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego
18. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach Science Publishers
19. Lax P.D., and Wendroff B. Difference schemes for hyperbolic equations with high order of accuracy. Commun. Pure Appl. Math. 17 (1964) 381-398
20. Sousa E. The controversial stability analysis. Appl. Math. Comput. 145 (2003) 777-794
21. Morton K.W. Numerical Solution of Convection Diffusion Problems (1996), Chapman and Hall, London
22. Warming F.F., and Hyett B.J. The modified equation approach to the stability and accuracy analysis of finite difference methods. J. Comput. Phys. 14 (1974) 159-179
23. Sod G.A. Numerical Methods in Fluid Dynamics: Initial and Initial Boundary-value Problems (1988), Cambridge University Press, Cambridge
24. McCulloch J.H., and Panton D.P. Precise tabulation of the maximally-skewed stable distributions and densities. Comput. Stat. Data Anal. 23 (1997) 307-320
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 vols. I and II (1990), Wiley-Interscience, Chichester
27. Smith G.D. Numerical Solution of Partial Differential Equations (1985), Oxford University Press, Oxford