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Volume 9, Issue 1
A Preconditioned Fast Finite Volume Method for Distributed-Order Diffusion Equation and Applications

Hongfei Fu, Huan Liu & Xiangcheng Zheng

East Asian J. Appl. Math., 9 (2019), pp. 28-44.

Published online: 2019-01

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  • Abstract

A Crank-Nicolson finite volume scheme for the modeling of the Riesz space distributed-order diffusion equation is proposed. The corresponding linear system has a symmetric positive definite Toeplitz matrix. It can be efficiently stored in  $\mathcal{O}$($NK$) memory. Moreover, for the finite volume scheme, a fast version of conjugate gradient (FCG) method is developed. Compared with the Gaussian elimination method, the computational complexity is reduced from $\mathcal{O}$($MN$3 + $NK$) to $\mathcal{O}$($l$$A$$MN$log$N$ + $NK$), where $l$$A$ is the average number of iterations at a time level. Further reduction of the computational cost is achieved due to use of a circulant preconditioner. The preconditioned fast finite volume method is combined with the Levenberg-Marquardt method to identify the free parameters of a distribution function. Numerical experiments show the efficiency of the method.

  • AMS Subject Headings

35R11, 65F08, 65F10, 65M08, 65T50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-9-28, author = {}, title = {A Preconditioned Fast Finite Volume Method for Distributed-Order Diffusion Equation and Applications}, journal = {East Asian Journal on Applied Mathematics}, year = {2019}, volume = {9}, number = {1}, pages = {28--44}, abstract = {

A Crank-Nicolson finite volume scheme for the modeling of the Riesz space distributed-order diffusion equation is proposed. The corresponding linear system has a symmetric positive definite Toeplitz matrix. It can be efficiently stored in  $\mathcal{O}$($NK$) memory. Moreover, for the finite volume scheme, a fast version of conjugate gradient (FCG) method is developed. Compared with the Gaussian elimination method, the computational complexity is reduced from $\mathcal{O}$($MN$3 + $NK$) to $\mathcal{O}$($l$$A$$MN$log$N$ + $NK$), where $l$$A$ is the average number of iterations at a time level. Further reduction of the computational cost is achieved due to use of a circulant preconditioner. The preconditioned fast finite volume method is combined with the Levenberg-Marquardt method to identify the free parameters of a distribution function. Numerical experiments show the efficiency of the method.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.160418.190518}, url = {http://global-sci.org/intro/article_detail/eajam/12933.html} }
TY - JOUR T1 - A Preconditioned Fast Finite Volume Method for Distributed-Order Diffusion Equation and Applications JO - East Asian Journal on Applied Mathematics VL - 1 SP - 28 EP - 44 PY - 2019 DA - 2019/01 SN - 9 DO - http://doi.org/10.4208/eajam.160418.190518 UR - https://global-sci.org/intro/article_detail/eajam/12933.html KW - Distributed-order diffusion equation, finite volume method, fast conjugate gradient method, circulant preconditioner, parameter identification. AB -

A Crank-Nicolson finite volume scheme for the modeling of the Riesz space distributed-order diffusion equation is proposed. The corresponding linear system has a symmetric positive definite Toeplitz matrix. It can be efficiently stored in  $\mathcal{O}$($NK$) memory. Moreover, for the finite volume scheme, a fast version of conjugate gradient (FCG) method is developed. Compared with the Gaussian elimination method, the computational complexity is reduced from $\mathcal{O}$($MN$3 + $NK$) to $\mathcal{O}$($l$$A$$MN$log$N$ + $NK$), where $l$$A$ is the average number of iterations at a time level. Further reduction of the computational cost is achieved due to use of a circulant preconditioner. The preconditioned fast finite volume method is combined with the Levenberg-Marquardt method to identify the free parameters of a distribution function. Numerical experiments show the efficiency of the method.

Hongfei Fu, Huan Liu & Xiangcheng Zheng. (2020). A Preconditioned Fast Finite Volume Method for Distributed-Order Diffusion Equation and Applications. East Asian Journal on Applied Mathematics. 9 (1). 28-44. doi:10.4208/eajam.160418.190518
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