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 θ (N K) 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 θ (MN3 + N K) to O (lAMN logN + N K), where lAis 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.

  • Keywords

Distributed-order diffusion equation finite volume method fast conjugate gradient method circulant preconditioner parameter identification.

  • AMS Subject Headings

35R11 65F08 65F10 65M08 65T50

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-9-28, author = {Hongfei Fu, Huan Liu and Xiangcheng Zheng}, 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 θ (N K) 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 θ (MN3 + N K) to O (lAMN logN + N K), where lAis 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} }
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