A Fast Preconditioning Strategy for QSC-CN Scheme of Space Fractional Diffusion Equations and Its Spectral Analysis (Advances in Applied Mathematics and Mechanics, 2024)
A quadratic spline collocation method combined with the Crank-Nicolsontime 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 theToeplitz-like matrix as the sum of a Toeplitz matrix and a rank-2 matrix and Strang’scirculant preconditioner is constructed to accelerate the convergence of Krylov subspace method like generalized minimal residual method for solving the discrete linearsystems. 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 proposedpreconditioner is better than that of the generalized T. Chan’s circulant preconditionerproposed 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 spacefractional diffusion equations.