Finite difference methods for the time fractional diffusion equation on non-uniform meshes

Journal of Computational Physics

Volumn 265, Issue , 2014, Pages 195-210

  Zhang, Ya nan ^a   Sun, Zhi zhong ^b   Liao, Hong lin ^c  

^a SOOCHOW UNIVERSITY   (China)

^b SOUTHEAST UNIVERSITY   (China)

^c PLA UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

Author keywords

Convergence; Finite difference method; Fractional diffusion equation; Non uniform meshes; Stability

Indexed keywords

DIFFUSION; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS;

CONVERGENCE; DISCRETIZATIONS; FINITE DIFFERENCE APPROXIMATIONS; FINITE-DIFFERENCE METHODS; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFUSION EQUATION; NON-UNIFORM MESH; TIME FRACTIONAL DIFFUSION EQUATION; UNIFORM MESH; WEAKLY SINGULAR KERNEL;

FINITE DIFFERENCE METHOD;

EID: 84894483339     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2014.02.008     Document Type: Article

References (40)

1. Oldham K.B., Spanier J. The Fractional Calculus 1974, Academic Press, New York.
2. Podlubny I. Fractional Differential Equations 1999, Academic Press, New York.
3. Bouchaud J., Georges A. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 1990, 195:127-293.
4. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 2000, 339:1-77.
5. Solomon T.H., Weeks E.R., Swinney H.L. Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow. Phys. Rev. Lett. 1993, 71:3975-3979.
6. Wyss W. Fractional diffusion equation. J. Math. Phys. 1986, 27:2782-2785.
7. Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 1996, 9:23-28.
8. Gorenflo R., Mainardi F., Moretti D., Paradisi P. Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 2002, 29:129-143.
9. Agrawal O.P. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 2002, 29:145-155.
10. Lubich Ch. Discretized fractional calculus. SIAM J. Math. Anal. 1986, 17:704-719.
11. Lubich Ch. Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput. 1985, 45:463-469.
12. Sanz-Serna J.M. A numerical method for a partial integro-differential equation. SIAM J. Numer. Anal. 1988, 25:319-327.
13. Lopez-Marcos J.C. A difference scheme for a nonlinear partial integrodifferential equation. SIAM J. Numer. Anal. 1990, 27:20-31.
14. Chen C., Thomée V., Wahlbin L.B. Finite element approximation of a parabolic integro-differential equation with weakly singular kernel. Math. Comput. 1992, 58:587-602.
15. Tang T. A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math. 1993, 11:309-319.
16. Cuesta E., Lubich C., Palencia C. Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 2006, 75:673-696.
17. Zhuang P., Liu F. Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 2006, 22:87-99.
18. Liu F., Zhuang P., Anh V., Turner I. A fractional-order implicit difference approximation for the space-time fractional diffusion equation. ANZIAM J. 2005, 47:C48-C68.
19. Sun Z.Z., Wu X.N. A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 2006, 56:193-209.
20. Du R., Cao W.R., Sun Z.Z. A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 2010, 34:2998-3007.
21. Brunner H., Ling L., Yamamoto M. Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 2010, 229:6613-6622.
22. Zhang Y.N., Sun Z.Z. Alternating direction implicit schemes for the two-dimensional fractional subdiffusion equation. J. Comput. Phys. 2011, 230:8713-8728.
23. Zhang Y.N., Sun Z.Z., Zhao X. Compact ADI schemes for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 2012, 50:1535-1555.
24. Deng W.H. Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 2008, 47:204-226.
25. Zhao X., Sun Z.Z. A box-type scheme for fractional subdiffusion equation with Neumann boundary conditions. J. Comput. Phys. 2011, 230:6061-6074.
26. Gao G.H., Sun Z.Z. A compact difference scheme for the fractional subdiffusion equations. J. Comput. Phys. 2011, 230:586-595.
27. Langlands T.A.M., Henry B.I. The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 2005, 205:719-736.
28. Lin Y., Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 2007, 225:1533-1552.
29. Lin Y., Li X., Xu C. Finite difference/spectral approximations for the fractional cable equation. Math. Comput. 2011, 80:1369-1396.
30. Zhang Y.N., Sun Z.Z., Wu H.W. Error estimates of Crank-Nicolson type difference schemes for the subdiffusion equation. SIAM J. Numer. Anal. 2011, 49:2302-2322.
31. Li X., Xu C. A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 2009, 47:2108-2131.
32. Chen C.M., Liu F., Anh V., Turner I. Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 2010, 32:1740-1760.
33. Brunner H. The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comput. 1985, 45:417-437.
34. Tang T. A note on collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J. Numer. Anal. 1993, 13:93-99.
35. Ma J., Jiang Y. On a graded mesh method for a class of weakly singular Volterra integral equations. J. Comput. Appl. Math. 2009, 231:807-814.
36. Mustapha K. An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements. IMA J. Numer. Anal. 2011, 31:719-739.
37. Yuste S.B., Quintana-Murillo J. A finite difference scheme with non-uniform timesteps for fractional diffusion equations. Comput. Phys. Commun. 2012, 183:2594-2600.
38. Samarskii A.A., Andreev V.B. Difference Methods for Elliptic Equation 1976, Nauka, Moscow.
39. Liao H.L., Sun Z.Z., Shi H.S. Error estimate of fourth-order compact scheme for linear Schrodinger equations. SIAM J. Numer. Anal. 2010, 47:4381-4401.
40. Sun Z.Z. The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations 2009, Science Press, Beijing.