An Efficient Preconditioner for Time–Space Fractional Diffusion Equations with Variable Coefficients
- 1School of Mathematics and Statistics, Beihua University, Jilin 132013, China
Abstract
This paper proposes a new class of efficient preconditioners for variable-coefficient time–space fractional diffusion equations. The Caputo time-fractional derivative is discretized by the L2-1σ scheme, while the Riesz space-fractional derivative is approximated by the shifted Grünwald–Letnikov formula, yielding an all-at-once system amenable to parallel solution. Because the coefficient matrix A of the resulting discrete system no longer has a strict Toeplitz structure, solving the system entails substantial storage and computational costs. To address this difficulty, we construct a class of Strang–τ structured preconditioners. The variable-coefficient diagonal matrix is first approximated using the arithmetic mean of the diffusion coefficient; the temporal and spatial Toeplitz matrices are then approximated by a Strang circulant matrix and a τ matrix, respectively. The preconditioner is applied efficiently using the fast Fourier transform (FFT), the discrete sine transform (DST), and the Sherman–Morrison rank-one update formula. We further derive a spectral distribution estimate showing that the eigenvalues of the preconditioned coefficient matrix P˜−1A lie in a disk centered at 1, and its radius is controlled jointly by the temporal low-rank correction and circulant-approximation errors, the spatial τ-approximation error, and the mean-approximation error of the variable coefficient. Numerical experiments demonstrate that the proposed preconditioner substantially accelerates GMRES(m) convergence.
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BibTeX
@article{zhao2026efficient,
title = {{An Efficient Preconditioner for Time–Space Fractional Diffusion Equations with Variable Coefficients}},
author = {Jianyu Zhao and Meijuan Wang and Yi Wang and Haifeng Sang},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213957},
url = {https://www.mdpi.com/2227-7390/14/21/3957},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - An Efficient Preconditioner for Time–Space Fractional Diffusion Equations with Variable Coefficients
AU - Jianyu Zhao
AU - Meijuan Wang
AU - Yi Wang
AU - Haifeng Sang
PY - 2026
DA - 2026/10/29
JO - Mathematics
DO - 10.3390/math14213957
UR - https://www.mdpi.com/2227-7390/14/21/3957
AB - This paper proposes a new class of efficient preconditioners for variable-coefficient time–space fractional diffusion equations. The Caputo time-fractional derivative is discretized by the L2-1σ scheme, while the Riesz space-fractional derivative is approximated by the shifted Grünwald–Letnikov formula, yielding an all-at-once system amenable to parallel solution. Because the coefficient matrix A of the resulting discrete system no longer has a strict Toeplitz structure, solving the system entails substantial storage and computational costs. To address this difficulty, we construct a class of Strang–τ structured preconditioners. The variable-coefficient diagonal matrix is first approximated using the arithmetic mean of the diffusion coefficient; the temporal and spatial Toeplitz matrices are then approximated by a Strang circulant matrix and a τ matrix, respectively. The preconditioner is applied efficiently using the fast Fourier transform (FFT), the discrete sine transform (DST), and the Sherman–Morrison rank-one update formula. We further derive a spectral distribution estimate showing that the eigenvalues of the preconditioned coefficient matrix P˜−1A lie in a disk centered at 1, and its radius is controlled jointly by the temporal low-rank correction and circulant-approximation errors, the spatial τ-approximation error, and the mean-approximation error of the variable coefficient. Numerical experiments demonstrate that the proposed preconditioner substantially accelerates GMRES(m) convergence.
SN - 2227-7390
ER -