TY - JOUR
T1 - A High-Order Accuracy Method for Solving the Fractional Diffusion Equations
AU - Ran , Maohua
AU - Zhang , Chengjian
JO - Journal of Computational Mathematics
VL - 2
SP - 239
EP - 253
PY - 2020
DA - 2020/02
SN - 38
DO - http://doi.org/10.4208/jcm.1805-m2017-0081
UR - https://global-sci.org/intro/article_detail/jcm/14516.html
KW - Boundary value method, Circulant preconditioner, High accuracy, Generalized Dirichlet type boundary condition.
AB - <p style="text-align: justify;">In this paper, an efficient numerical method for solving the general fractional diffusion
equations with Riesz fractional derivative is proposed by combining the fractional compact
difference operator and the boundary value methods. In order to efficiently solve the
generated linear large-scale system, the generalized minimal residual (GMRES) algorithm
is applied. For accelerating the convergence rate of the iterative, the Strang-type, Chan-type and P-type preconditioners are introduced. The suggested method can reach higher
order accuracy both in space and in time than the existing methods. When the used
boundary value method is $A_{k1,k2}$-stable, it is proven that Strang-type preconditioner is
invertible and the spectra of preconditioned matrix is clustered around 1. It implies that
the iterative solution is convergent rapidly. Numerical experiments with the absorbing
boundary condition and the generalized Dirichlet type further verify the efficiency.</p>