More Web Proxy on the site http://driver.im/

article

Compact finite difference method for the fractional diffusion equation

Author:

Mingrong CuiAuthors Info & Claims

Journal of Computational Physics, Volume 228, Issue 20

Pages 7792 - 7804

https://doi.org/10.1016/j.jcp.2009.07.021

Published: 01 November 2009 Publication History

Abstract

High-order compact finite difference scheme for solving one-dimensional fractional diffusion equation is considered in this paper. After approximating the second-order derivative with respect to space by the compact finite difference, we use the Grunwald-Letnikov discretization of the Riemann-Liouville derivative to obtain a fully discrete implicit scheme. We analyze the local truncation error and discuss the stability using the Fourier method, then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.

References

[1]

Metzler, R. and Klafter, J., The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. v339. 1-77.

[2]

Podlubny, I., Fractional Differential Equations. 1999. Academic Press, San Diego.

[3]

Lynch, V.E., Carreras, B.A., Del-Castillo-Negrete, D., Ferreira-Mejias, K.M. and Hicks, H.R., Numerical methods for the solution of partial differential equations of fractional order. J. Comput. Phys. v192. 406-421.

Digital Library

[4]

Meerschaert, M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. v172. 65-77.

Digital Library

[5]

Langlands, T.A.M. and Henry, B.I., The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. v205. 719-736.

Digital Library

[6]

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. v42 i5. 1862-1874.

[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. v227. 886-897.

[8]

Zhuang, P., Liu, F., Anh, V. and Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. v46 i2. 1079-1095.

[9]

Chen, C.-M., Liu, F. and Burrage, K., Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. v198. 754-769.

[10]

Liu, F., Yang, C. and Burrage, K., Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. v231. 160-176.

Digital Library

[11]

Scherer, R., Kalla, S.L., Boyadjiev, L. and Al-Saqabi, B., Numerical treatment of fractional heat equations. Appl. Numer. Math. v58. 1212-1223.

Digital Library

[12]

Podlubny, I., Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. v3 i4. 359-386.

[13]

Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y. and Vinagre Jara, B.M., Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. v228. 3137-3153.

Digital Library

[14]

Xu, M.Y. and Tan, W.C., Theoretical analysis of the velocity field stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion. Sci. China Ser. A: Math. v44. 1387-1399.

[15]

Hirsch, R.S., Higher order accurate difference solutions of fluid mechanics problems by a compact difference technique. J. Comput. Phys. v24. 90-109.

[16]

Lele, S.K., Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. v103. 16-42.

[17]

Liao, W., Zhu, J. and Khaliq, A.Q.M., An efficient high-order algorithm for solving systems of reaction-diffusion equations. Numer. Meth. Partial Diff. Eq. v18. 340-354.

[18]

Cui, M.R., Fourth-order compact scheme for the one-dimensional sine-Gordon equation. Numer. Meth. Partial Diff. Eq. v25. 685-711.

[19]

Zhao, J.C., Zhang, T. and Corless, R.M., Convergence of the compact finite difference method for second-order elliptic equations. Appl. Math. Comput. v182. 1454-1469.

[20]

Ames, W.F., Numerical Methods for Partial Differential Equations. 1977. Academic Press, New York.

[21]

Lubich, C.H., Discretized fractional calculus. SIAM J. Math. Anal. v17 i3. 704-719.

[22]

Thomas, W., . 1995. Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 1995.Springer, -Verlag.

[23]

A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics 23, Springer-Verlag, 1994.

[24]

Horn, Roger A. and Johnson, Charles R., Matrix Analysis. 1986. Cambridge University Press.

[25]

Tadjeran, Charles and Meerschaert, Mark M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. v220. 813-823.

Digital Library

[26]

Lin, Y.M. and Xu, C.J., Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. v225. 1533-1552.

Digital Library

Cited By

Zuffi GLobato FCavallini ASteffen V(2024)Numerical solution and sensitivity analysis of time–space fractional near-field acoustic levitation model using Caputo and Grünwald–Letnikov derivativesSoft Computing - A Fusion of Foundations, Methodologies and Applications10.1007/s00500-024-09757-128:13-14(8457-8470)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s00500-024-09757-1
Pu TFasondini M(2023)The numerical solution of fractional integral equations via orthogonal polynomials in fractional powersAdvances in Computational Mathematics10.1007/s10444-022-10009-949:1Online publication date: 4-Feb-2023
https://dl.acm.org/doi/10.1007/s10444-022-10009-9
Maurya RSingh V(2022)A high-order adaptive numerical algorithm for fractional diffusion wave equation on non-uniform meshesNumerical Algorithms10.1007/s11075-022-01372-192:3(1905-1950)Online publication date: 5-Aug-2022
https://dl.acm.org/doi/10.1007/s11075-022-01372-1
Show More Cited By

Compact finite difference method for the fractional diffusion equation
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations

Recommendations

Compact alternating direction implicit method for two-dimensional time fractional diffusion equation

High-order compact finite difference scheme with operator splitting technique for solving two-dimensional time fractional diffusion equation is considered in this paper. A Grunwald-Letnikov approximation is used for the Riemann-Liouville time derivative,...
Finite difference/spectral approximations for the time-fractional diffusion equation

In this paper, we consider the numerical resolution of a time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative (of order @a, with 0=<@a=<1). ...
A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection---diffusion equations

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection---diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Computational Physics

Journal of Computational Physics Volume 228, Issue 20

November, 2009

368 pages

ISSN:0021-9991

Issue’s Table of Contents

Copyright © Elsevier Inc. © 2009.

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 01 November 2009

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

86
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 24 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Zuffi GLobato FCavallini ASteffen V(2024)Numerical solution and sensitivity analysis of time–space fractional near-field acoustic levitation model using Caputo and Grünwald–Letnikov derivativesSoft Computing - A Fusion of Foundations, Methodologies and Applications10.1007/s00500-024-09757-128:13-14(8457-8470)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s00500-024-09757-1
Pu TFasondini M(2023)The numerical solution of fractional integral equations via orthogonal polynomials in fractional powersAdvances in Computational Mathematics10.1007/s10444-022-10009-949:1Online publication date: 4-Feb-2023
https://dl.acm.org/doi/10.1007/s10444-022-10009-9
Maurya RSingh V(2022)A high-order adaptive numerical algorithm for fractional diffusion wave equation on non-uniform meshesNumerical Algorithms10.1007/s11075-022-01372-192:3(1905-1950)Online publication date: 5-Aug-2022
https://dl.acm.org/doi/10.1007/s11075-022-01372-1
Lin J(2022)Simulation of 2D and 3D inverse source problems of nonlinear time-fractional wave equation by the meshless homogenization function methodEngineering with Computers10.1007/s00366-021-01489-238:Suppl 4(3599-3608)Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s00366-021-01489-2
Tian XReutskiy SFu Z(2022)A novel meshless collocation solver for solving multi-term variable-order time fractional PDEsEngineering with Computers10.1007/s00366-021-01298-738:Suppl 2(1527-1538)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s00366-021-01298-7
Zheng XErvin VWang H(2021)Optimal Petrov–Galerkin Spectral Approximation Method for the Fractional Diffusion, Advection, Reaction Equation on a Bounded IntervalJournal of Scientific Computing10.1007/s10915-020-01366-y86:3Online publication date: 1-Mar-2021
https://dl.acm.org/doi/10.1007/s10915-020-01366-y
Safari FSun H(2021)Improved singular boundary method and dual reciprocity method for fractional derivative Rayleigh–Stokes problemEngineering with Computers10.1007/s00366-020-00991-337:4(3151-3166)Online publication date: 1-Oct-2021
https://dl.acm.org/doi/10.1007/s00366-020-00991-3
Castillo PGómez S(2020)On the Convergence of the Local Discontinuous Galerkin Method Applied to a Stationary One Dimensional Fractional Diffusion ProblemJournal of Scientific Computing10.1007/s10915-020-01335-585:2Online publication date: 22-Oct-2020
https://dl.acm.org/doi/10.1007/s10915-020-01335-5
Zureigat HIsmail ASathasivam S(2020)A compact Crank–Nicholson scheme for the numerical solution of fuzzy time fractional diffusion equationsNeural Computing and Applications10.1007/s00521-019-04148-232:10(6405-6412)Online publication date: 1-May-2020
https://dl.acm.org/doi/10.1007/s00521-019-04148-2
Lyu PVong S(2018)A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equationsNumerical Algorithms10.1007/s11075-017-0385-y78:2(485-511)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s11075-017-0385-y
Show More Cited By

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents