TY - JOUR
T1 - A Separable Preconditioner for Time-Space Fractional Caputo-Riesz Diffusion Equations
AU - Xuelei Lin, Michael K. Ng & Haiwei Sun
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 827
EP - 853
PY - 2018
DA - 2018/06
SN - 11
DO - http://doi.org/10.4208/nmtma.2018.s09
UR - https://global-sci.org/intro/article_detail/nmtma/12475.html
KW - 
AB - <p style="text-align: justify;">In this paper, we study linear systems arising from time-space fractional
Caputo-Riesz diffusion equations with time-dependent diffusion coefficients. The coefficient
matrix is a summation of a block-lower-triangular-Toeplitz matrix (temporal component)
and a block-diagonal-with-diagonal-times-Toeplitz-block matrix (spatial component).
The main aim of this paper is to propose separable preconditioners for solving
these linear systems, where a block ϵ-circulant preconditioner is used for the temporal
component, while a block diagonal approximation is used for the spatial variable. The
resulting preconditioner can be block-diagonalized in the temporal domain. Furthermore,
the fast solvers can be employed to solve smaller linear systems in the spatial
domain. Theoretically, we show that if the diffusion coefficient (temporal-dependent
or spatial-dependent only) function is smooth enough, the singular values of the preconditioned
matrix are bounded independent of discretization parameters. Numerical
examples are tested to show the performance of proposed preconditioner.</p>