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.