An Efficient Preconditioner for Time–Space Fractional Diffusion Equations with Variable Coefficients

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.