East Asian J. Appl. Math., 9 (2019), pp. 28-44.
Published online: 2019-01
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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$$A$ is the average number of iterations at a time level. Further reduction of the computational cost is achieved due to use of a circulant preconditioner. The preconditioned fast finite volume method is combined with the Levenberg-Marquardt method to identify the free parameters of a distribution function. Numerical experiments show the efficiency of the method.
}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.160418.190518}, url = {http://global-sci.org/intro/article_detail/eajam/12933.html} }A Crank-Nicolson finite volume scheme for the modeling of the Riesz space distributed-order diffusion equation is proposed. The corresponding linear system has a symmetric positive definite Toeplitz matrix. It can be efficiently stored in $\mathcal{O}$($NK$) memory. Moreover, for the finite volume scheme, a fast version of conjugate gradient (FCG) method is developed. Compared with the Gaussian elimination method, the computational complexity is reduced from $\mathcal{O}$($MN$3 + $NK$) to $\mathcal{O}$($l$$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.