TY - JOUR
T1 - A Fast Preconditioning Strategy for QSC-CN Scheme of Space Fractional Diffusion Equations and Its Spectral Analysis
AU - Qu , Wei
AU - Huang , Yuanyuan
AU - Lei , Siu-Long
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1474
EP - 1501
PY - 2024
DA - 2024/10
SN - 16
DO - http://doi.org/10.4208/aamm.OA-2023-0050
UR - https://global-sci.org/intro/article_detail/aamm/23475.html
KW - Circulant preconditioning, Toeplitz-like matrix, matrix splitting, spectral analysis, Krylov subspace iterative method.
AB - <p style="text-align: justify;">A quadratic spline collocation method combined with the Crank-Nicolson
time discretization of the space fractional diffusion equations gives discrete linear systems, whose coefficient matrix is the sum of a tridiagonal matrix and two diagonal-multiply-Toeplitz-like matrices. By exploiting the Toeplitz-like structure, we split the
Toeplitz-like matrix as the sum of a Toeplitz matrix and a rank-2 matrix and Strang’s
circulant preconditioner is constructed to accelerate the convergence of Krylov subspace method like generalized minimal residual method for solving the discrete linear
systems. In theory, both the invertibility of the proposed preconditioner and the clustering spectrum of the corresponding preconditioned matrix are discussed in detail. Finally, numerical results are given to demonstrate that the performance of the proposed
preconditioner is better than that of the generalized T. Chan’s circulant preconditioner
proposed recently by Liu et al. (J. Comput. Appl. Math., 360 (2019), pp. 138–156)
for solving the discrete linear systems of one-dimensional and two-dimensional space
fractional diffusion equations.</p>