Banded preconditioning with shift compensation for solving discrete Riesz space-fractional diffusion equations

Based on the finite-difference method, the considered Riesz space-fractional diffusion equations result in a series of linear systems, whose coefficient matrices are composed of the identity matrix and the product of diagonal matrix and Toeplitz matrix. With the aid of the Toeplitz structure contained in the discrete linear system, a class of banded preconditioner with shift compensation (BSC preconditioner) is designed to improve the convergence rates of the Krylov subspace iteration methods. The BSC preconditioner can be viewed as a modification of the banded preconditioner with diagonal compensation (BDC preconditioner) proposed by F.-R. Lin et al. (F.-R. Lin, S.-W. Yang and X.-Q. Jin, J. Comput. Phys. 256, 109–117 2014), but unlike the BDC preconditioner, it retains the same Toeplitz-like structure as the coefficient matrix of the discrete linear system. Although the theoretical result on the eigenvalues of the BDC-preconditioned matrix has not been given so far, the eigenvalue distributions of the BSC-preconditioned matrix and the BDC-preconditioned matrix can be demonstrated simultaneously by using the structure of the BSC preconditioner. In addition, theoretical analysis shows that the eigenvalues of the BSC-preconditioned matrix are real and located in a positive interval \([1,\nu )\) $[1,\nu )$ with \(\nu \) $\nu$ weakly dependent on h, while the eigenvalues of the BDC-preconditioned matrix are weakly clustered around those of the BSC-preconditioned matrix, which indicates that the BSC preconditioner is almost as effective as the BDC preconditioner. Numerical experiments reveal that both the BSC preconditioner and BDC preconditioner can significantly accelerate the convergence rates of the Krylov subspace iteration methods, and can show h-independent convergence behavior when the order of the fractional derivative approaches 2.