A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation

Journal of Computational Physics

Volumn 253, Issue , 2013, Pages 50-63

  Wang, Hong ^a   Du, Ning ^b  

^a UNIVERSITY OF SOUTH CAROLINA   (United States)

^b SHANDONG UNIVERSITY   (China)

Author keywords

Anomalous diffusion; Circulant matrix; Conjugate gradient squared method; Fast Fourier transform; Space fractional diffusion equation; Toeplitz matrix

Indexed keywords

CONJUGATE GRADIENT METHOD; DIFFUSION; FAST FOURIER TRANSFORMS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS;

ANOMALOUS DIFFUSION; CIRCULANT MATRIX; COEFFICIENT MATRIX; COMPUTATIONAL WORK; CONJUGATE GRADIENT SQUARED METHOD; FINITE-DIFFERENCE METHODS; FRACTIONAL DIFFUSION EQUATION; SPACE-FRACTIONAL DIFFUSION EQUATION; SPACE-FRACTIONAL DIFFUSIONS; TOEPLITZ MATRICES;

FINITE DIFFERENCE METHOD;

EID: 84880978495     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2013.06.040     Document Type: Article

References (40)

1. Barrett R., Berry M., Chan T.F., Demmel J., Donato J.M., Dongarra J., Eijkhout V., Pozo R., Romine C., Van der Vorst H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods 1994, SIAM, Philadelphia, PA.
2. Basu T.S., Wang H. A fast second-order finite difference method for space-fractional diffusion equations. Int'l J. Numer. Anal. Modeling 2012, 9:658-666.
3. Benson D., Wheatcraft S.W., Meerschaert M.M. The fractional-order governing equation of Lévy motion. Water Resour. Res. 2000, 36:1413-1423.
4. Chen S., Liu F. ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation. J. Appl. Math. Comput. 2008, 26:295-311.
5. Davis P.J. Circulant Matrices 1979, Wiley-Interscience, New York.
6. Ervin V.J., Heuer N., Roop J.P. Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 2007, 45:572-591.
7. Ervin V.J., Roop J.P. Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differential Equ. 2005, 22:558-576.
8. R d. Numer. Methods Partial Differential Equ. 2007, 23:256-281.
9. Gray R.M. Toeplitz and circulant matrices: A review. Found. Trends Commun. Inf. Theory 2006, 2(3):55-239.
10. Langlands T.A.M., Henry B.I., Wearne S.L. Fractional cable equation models for anomalous electrodiffusion in nerve cells: Infinite domain solutions. J. Math. Biol. 2009, 59:761-808.
11. Li X., Xu C. The existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 2010, 8:1016-1051.
12. Li C., Zeng F., Liu F. Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 2012, 15:383-406.
13. Lin R., Liu F., Anh V., Turner I. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 2009, 212:435-445.
14. Liu F., Anh V., Turner I. Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 2004, 166:209-219.
15. Liu Q., Liu F., Turner I., Anh V. Numerical simulation for the three-dimensional seepage flow with fractional derivatives in porous media. IMA J. Appl. Math. 2009, 74:201-229.
16. Liu Q., Liu F., Turner I., Anh V. Finite element approximation for the modified anomalous subdiffusion process. Appl. Math. Model. 2011, 35:4103-4116.
17. Mainardi F. Fractals and Fractional Calculus Continuum Mechanics 1997, 291-348. Springer-Verlag.
18. Meerschaert M.M., Scheffler H.P., Tadjeran C. Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 2006, 211:249-261.
19. Meerschaert M.M., Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 2004, 172:65-77.
20. Meerschaert M.M., Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 2006, 56:80-90.
21. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 2000, 339:1-77.
22. Murio D.A. Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 2008, 56:1138-1145.
23. Oldham K.B. Fractional differential equations in electrochemistry. Adv. Engrg. Software 2010, 41:9-12.
24. Podlubny I. Fractional Differential Equations 1999, Academic Press, New York.
25. Raberto M., Scalas E., Mainardi F. Waiting-times and returns in high-frequency financial data: An empirical study. Physica 2002, 314:749-755.
26. Ramos-Fernandez G., Mateos J., Miramontes O., Cocho G., Larralde H., Ayala-Orozco B. Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi). Behav. Ecol. Sociobiol. 2004, 55:223-230.
27. Ritchie K., Shan X.-Y., Kondo J., Iwasawa K., Fujiwara T., Kusumi A. Detection of non-Brownian diffusion in the cell membrane in single molecule tracking. Biophysical J. 2005, 88:2266-2277.
28. R2. J. Comput. Appl. Math. 2006, 193:243-268.
29. Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications 1993, Gordon and Breach, Yverdon.
30. Scher H., Montroll E.W. Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 1975, 12:2455-2477.
31. Sousa E. Finite difference approximates for a fractional advection diffusion problem. J. Comput. Phys. 2009, 228:4038-4054.
32. Su L., Wang W., Wang H. A characteristic finite difference method for transient fractional advection-diffusion equations. Appl. Numer. Math. 2011, 61:946-960.
33. Varga R.S. Matrix Iterative Analysis 2000, Springer-Verlag, Berlin, Heidelberg. second edition.
34. Wang H., Basu T. A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 2012, 34:A2444-A2458.
35. Wang H., Du N. A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations. J. Comput. Phys. 2013, 240:49-57.
36. log2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations. J. Comput. Phys. 2011, 230:7830-7839.
37. log2N) finite difference method for fractional diffusion equations. J. Comput. Phys. 2010, 229:8095-8104.
38. Wang K., Wang H. A fast characteristic finite difference method for fractional advection-diffusion equations. Adv. Water Resour. 2011, 34:810-816.
39. Wang H., Yang D. Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 2013, 51:1088-1107.
40. Yu Q., Liu F., Turner I., Burrage K. A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D. Appl. Math. Comp. 2012, 219:4082-4095.