Fast permutation preconditioning for fractional diffusion equations

In this paper, an implicit finite difference scheme with the shifted Grunwald formula, which is unconditionally stable, is used to discretize the fractional diffusion equations with constant diffusion coefficients. The coefficient matrix possesses the Toeplitz structure and the fast Toeplitz matrix-vector product can be utilized to reduce the computational complexity from \({\mathcal {O}}{(N^{3})}\) to \({\mathcal {O}}{(N \log N)}\), where N is the number of grid points. Two preconditioned iterative methods, named bi-conjugate gradient method for Toeplitz matrix and bi-conjugate residual method for Toeplitz matrix, are proposed to solve the relevant discretized systems. Finally, numerical experiments are reported to show the effectiveness of our preconditioners.