Fast solution algorithms for exponentially tempered fractional diffusion equations

Abstract

In this article, a fast‐iterative method and a fast‐direct method is proposed for solving one‐dimensional and two‐dimensional tempered fractional diffusion equations with constant coefficients. The proposed iterative method is accelerated by circulant preconditioning which is shown to converge superlinearly while the proposed direct method is based on circulant and skew‐circulant representation for Toeplitz matrix inversion. In one‐dimensional case, the operation cost of the proposed methods are both shown to be O ( N log N ) with O ( N ) memory requirement in each time step, where N is the number of spatial nodes. With the alternating direction implicit method, it is proven that the proposed fast solution algorithms can be extended to handle two‐dimensional tempered fractional diffusion equations with O ( N 2 log N ) operation cost and O ( N 2 ) memory requirement in each time step, where the number of spatial nodes in x ‐direction and y ‐direction both equal to N . Numerical examples are provided to illustrate the effectiveness and efficiency of the proposed methods.

A Preconditioning Technique for All-at-Once System from the Nonlinear Tempered Fractional Diffusion Equation

“…To address this problem, Wang et al [9] proposed a circulant preconditioned generalized minimal residual method (PGMRES) to solve the discretized linear system, whose computational cost is of O(N log N ). Lei et al [23] proposed fast solution algorithms for solving TFDEs in one-dimensional (1D) and two-dimensional (2D). In their article, for 1D case, a circulant preconditioned iterative method and a fast-direct method are developed, and the computational complexity of both methods are O(N log N ) in each time step.…”