In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of space-time fractional advection-diffusion equations. To start with, an implicit method based on two-sided Gr\"unwald formulae is proposed with a discussion of the stability and consistency. Then, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual ({preconditioned} CGNR) method, with an easily constructed preconditioner, are developed. Importantly, because the resulting systems are Topelitz-like, the fast Fourier transform can be applied to significantly reduce the computational cost. Numerical experiments are implemented to show the efficiency of our preconditioner, even with cases of variable coefficients.
@article{arxiv.1510.05089,
title = {Preconditioned iterative methods for space-time fractional advection-diffusion equations},
author = {Zhi Zhao and Xiao-Qing Jin and Matthew M. Lin},
journal= {arXiv preprint arXiv:1510.05089},
year = {2016}
}