Approximate Inverse Circulant Preconditioning Method For Fractional Advection-diffusion Equations

Recently, fractional advection-diffusion equations have been playing more and more important roles in physics, groundwater hydrology and so on. Because of the nonlocal properties of fractional operators, directly solving the exact solutions of the fractional advection-diffusion equations becomes very difficult. Therefore,numerically solving a class of the equations is the best choice. In this paper, we propose approximate inverse circulant preconditioning method for fractional advectiondiffusion equations with variable coefficients. Firstly, the equations are discretized by using the implicit finite difference scheme with the Gr¨unwald and shifted Gr¨unwald formulas, which results in nonsymmetry Toeplitz-like linear systems. The coefficient matrix can be written as the sum of an identity matrix and four diagonalmultiply-Toeplitz matrices. Secondly, we construct approximate inverse circulant preconditioners for such Toeplitz-like linear systems by using the inverse of weighted R. Chan’s circulant matrices and the interpolation method to approximate the inverse of Toeplitz-like matrices. Besides, we prove theoretically that each of the approximate inverse circulant preconditioned matrices can be written as the sum of an identity matrix, a low rank matrix and a small norm matrix, that is, the spectra of the preconditioned matrices are clustered around one. Finally, numerical results also demonstrate that our proposed approximate inverse circulant preconditioners are very effective.