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Volume 9, Issue 3
A Fast Second-Order Finite Difference Method for Space-Fractional Diffusion Equations

T. S. Basu & H. Wang

Int. J. Numer. Anal. Mod., 9 (2012), pp. 658-666.

Published online: 2012-09

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  • Abstract

Fractional diffusion equations provide an adequate and accurate description of transport processes that exhibit anomalous diffusion that cannot be modeled accurately by classical second-order diffusion equations. However, numerical discretizations of fractional diffusion equations yield full coefficient matrices, which require a computational operation of $O(N^3)$ per time step and a memory of $O(N^2)$ for a problem of size $N$. In this paper we develop a fast second-order finite difference method for space-fractional diffusion equations, which only requires memory of $O(N)$ and computational work of $O(N log^2 N)$. Numerical experiments show the utility of the method.

  • Keywords

circulant and Toeplitz matrix, fast direct solver, fast finite difference methods, fractional diffusion equations.

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-9-658, author = {Basu , T. S. and Wang , H.}, title = {A Fast Second-Order Finite Difference Method for Space-Fractional Diffusion Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2012}, volume = {9}, number = {3}, pages = {658--666}, abstract = {

Fractional diffusion equations provide an adequate and accurate description of transport processes that exhibit anomalous diffusion that cannot be modeled accurately by classical second-order diffusion equations. However, numerical discretizations of fractional diffusion equations yield full coefficient matrices, which require a computational operation of $O(N^3)$ per time step and a memory of $O(N^2)$ for a problem of size $N$. In this paper we develop a fast second-order finite difference method for space-fractional diffusion equations, which only requires memory of $O(N)$ and computational work of $O(N log^2 N)$. Numerical experiments show the utility of the method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/652.html} }
TY - JOUR T1 - A Fast Second-Order Finite Difference Method for Space-Fractional Diffusion Equations AU - Basu , T. S. AU - Wang , H. JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 658 EP - 666 PY - 2012 DA - 2012/09 SN - 9 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/652.html KW - circulant and Toeplitz matrix, fast direct solver, fast finite difference methods, fractional diffusion equations. AB -

Fractional diffusion equations provide an adequate and accurate description of transport processes that exhibit anomalous diffusion that cannot be modeled accurately by classical second-order diffusion equations. However, numerical discretizations of fractional diffusion equations yield full coefficient matrices, which require a computational operation of $O(N^3)$ per time step and a memory of $O(N^2)$ for a problem of size $N$. In this paper we develop a fast second-order finite difference method for space-fractional diffusion equations, which only requires memory of $O(N)$ and computational work of $O(N log^2 N)$. Numerical experiments show the utility of the method.

T. S. Basu & H. Wang. (1970). A Fast Second-Order Finite Difference Method for Space-Fractional Diffusion Equations. International Journal of Numerical Analysis and Modeling. 9 (3). 658-666. doi:
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