33 research outputs found
Fast iterative method with a second-order implicit difference scheme for time-space fractional convectionâdiffusion equation
In this paper we want to propose practical numerical methods to solve a class
of initial-boundary problem of time-space fractional convection-diffusion
equations (TSFCDEs). To start with, an implicit difference method based on
two-sided weighted shifted Gr\"{u}nwald formulae is proposed with a discussion
of the stability and convergence. We construct an implicit difference scheme
(IDS) and show that it converges with second order accuracy in both time and
space. Then, we develop fast solution methods for handling the resulting system
of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers
with suitable circulant preconditioners are designed to deal with the resulting
Toeplitz linear systems. Each time level of these methods reduces the memory
requirement of the proposed implicit difference scheme from
to and the computational complexity from to in each iterative step, where is the number of grid nodes. Extensive
numerical example runs show the utility of these methods over the traditional
direct solvers of the implicit difference methods, in terms of computational
cost and memory requirements.Comment: 29 pages, 8 tables, 4 figures. Submitted to academic journal for
peer-review. arXiv admin note: text overlap with arXiv:1510.05089,
arXiv:1503.04886 by other author