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 - <p style="text-align: justify;">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&nbsp; $\mathcal{O}$($NK$) 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&nbsp;$\mathcal{O}$($MN$<sup>3</sup> + $NK$) to $\mathcal{O}$($l$<sub>$A$</sub>$MN$log$N$ + $NK$), where $l$<sub>$A$</sub> 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.</p>