Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation

Numerical Algorithms

Volumn 54, Issue 1, 2010, Pages 1-21

  Chen, Chang Ming ^a   Liu, Fawang ^b   Turner, Ian ^b   Anh, Vo ^b  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Anomalous dynamics; Convergence; Fourier analysis; Multivariate extrapolation; Stability; Subdiffusion equation; The two dimensional problem

Indexed keywords

EID: 77950690888     PISSN: 10171398     EISSN: 15729265     Source Type: Journal    
DOI: 10.1007/s11075-009-9320-1     Document Type: Article

References (36)

1. Compte, A.: Continuous time random walks on moving fluids. Phys. Rew. E 55, 6821-6831 (1997).
2. Compte, A., Caceres, M. O.: Fractional dymamics in random velocity fields. Phys. Rev. Lett. 81, 3140-3143 (1998).
3. Henry, B. I., Wearne, S. L.: Fractional reaction-diffusion. Physica A 276, 448-455 (2000).
4. Baeumer, B., Meerschaert, M. M., Benson, D. A., Wheatsraft, S. W.: Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37, 1543-1550 (2001).
5. Chen, C.-M., Liu, F., Turner, I., Anh, V.: A new Fourier analysis method for the Galilei invariant fractional advection diffusion equation. ANZIAM J. 48(CTAC2006), C775-C789 (2007).
6. Chen, C.-M., Liu, F., Turner, I., Anh, V.: Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886-897 (2007).
7. Chen, C.-M., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. 198, 754-769 (2008).
8. Chen, C.-M., Liu, F., Anh, V.: Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Appl. Math. Comput. 204, 340-351 (2008).
9. Chen, C.-M., Liu, F., Anh, V.: A Fourier method and an extrapolation technique for Stokes first problem for a heated generalized second grade fluid with fractional derivative. J. Comput. Appl. Math. 223, 777-789 (2009).
10. Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673-696 (2006).
11. Enson, D. A., Wheatcraft, S. W., Meerschaert, M. M.: Application of a fractional advection-despersion equation. Water Resour. Res. 36(6), 1403-1412 (2000).
12. Ervin, V. S., Roop, J. P., Paul, J.: Variational solution of fractional advection dispersion equations on bounded domains in R-d. Numer. Meth. Part. D. E. 23, 256-281 (2007).
13. Podlubny, I.: Fractional Differential Equations. Academic, New York (1999).
14. Liu, F., Shen, S., Anh, V., Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. ANZIAM J. 46(E), 488-504 (2005).
15. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191, 12-20 (2007).
16. Cushman, J. H., Cinn, T. R.: Fractional advection-disperision equation: a classical mass balance with convolution-Fickian flux. Water Resour. Res. 36, 3763-3766 (2000).
17. 2. J. Comput. Appl. Math. 193, 243-268 (2006).
18. Seki, K., Wojcik, M., Tachiya, M.: Fractional reaction-diffusion equation. J. Chem. Phys. 119, 2165-2174 (2003).
19. Meerschaert, M. M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65-77 (2004).
20. Meerschaert, M. M., Nane, E., Xiao, Y.: Large deviations for local time fractional Brownian motion and applications. J. Math. Anal. Appl. 346, 432-445 (2008).
21. Giona, M., Roman, H. E.: Fractional diffusion equation for transport phenomena in randam media. Physica A 185, 87-97 (1992).
22. Zhuang, P., Liu, F.: Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22(3), 87-99 (2006).
23. Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation. SIAM J. Numer. Anal. 46(2), 1079-1095 (2008).
24. Metzler, R., Klafter, J., Sokolov, I. M.: Anomalous transport in external fields: continuous time random walksand fractional diffusion equations extended. Phys. Rev. E 58, 1621-1633 (1998).
25. Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, 3563-3567 (1999).
26. Metzler, R., Klafter, J.: The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1-77 (2000).
27. Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Physica A 278, 107-125 (2000).
28. Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284, 521-541 (2002).
29. Yuste, S. B., Acedo, L.: An explicit finite difference method and a new Von neumman-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862-1874 (2005).
30. Shen, S., Liu, F.: Error analysis of an explicit finite difference approximation for the space fractional diffusion. ANZIAM J. 46(E), 871-887 (2005).
31. Langlands, T. A. M., Henry, B. I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comp. Phys. 205, 719-736 (2005).
32. Balakrishnan, V.: Anomalous diffusion in one dimension. Physica A 132, 569-580 (1985).
33. McLean, W., Mustapha, K.: Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algor. (2009). doi: 10. 1007/s11075-008-9258-8.
34. Schneider, W. R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134-144 (1989).
35. Wyss, W.: Fractional diffusion equation. J. Math. Phys. 27, 2782-2785 (1986).
36. Zhang, Y., Meerschaert, M. M., Baeumer, B.: Particle tracking for time-fractional diffusion. Phys. Rev. E 78, 036705 (2008).