R. Chan’s circulant-based approximate inverse preconditioning iterative method for solving second-order space fractional advection–dispersion equations with variable coefficients

In this paper, we consider a second-order approximation scheme for the space fractional advection–diffusion equations with variable coefficients. This scheme employs the weighted and shifted Grünwald difference formulas to approximate the Riemann–Liouville fractional derivatives and the Crank–Nicols...

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Bibliographic Details
Published in:Computational & applied mathematics 2024-03, Vol.43 (2), Article 97
Main Authors: Tang, Shi-Ping, Yang, Ai-Li, Zhou, Jian-Lin, Wu, Yu-Jiang
Format: Article
Language:English
Subjects:
Advection
Advection-diffusion equation
Applications of Mathematics
Approximation
Coefficients
Computational Mathematics and Numerical Analysis
Interpolation
Iterative methods
Linear systems
Mathematical Applications in Computer Science
Mathematical Applications in the Physical Sciences
Mathematics
Mathematics and Statistics
Preconditioning
Robustness (mathematics)
Stability analysis
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Summary:In this paper, we consider a second-order approximation scheme for the space fractional advection–diffusion equations with variable coefficients. This scheme employs the weighted and shifted Grünwald difference formulas to approximate the Riemann–Liouville fractional derivatives and the Crank–Nicolson scheme for the temporal derivative, respectively. The stability and convergence of this approximation scheme are analyzed under certain conditions satisfied by the coefficients. This numerical scheme results in a sequence of nonsymmetric Toeplitz-like linear systems whose coefficient matrix can be written as a sum of an identity matrix and four diagonal-times-Toeplitz matrices. To improve the computational efficiency of the Krylov subspace iteration methods used to solve the scheme, we develop an approximate inverse preconditioner. This preconditioner is based on the inverse of the weighted R. Chan’s circulant matrix and the interpolation method. We also demonstrate that the preconditioned matrix can be written as a sum of an identity matrix, a low-rank matrix, and a small-norm matrix, resulting in a clustered spectrum around one. The numerical results demonstrate that the numerical scheme and its approximate inverse preconditioner are robust and efficient.
ISSN:2238-3603
1807-0302
DOI:10.1007/s40314-024-02592-y