Strang-type preconditioners for solving fractional diffusion equations by boundary value methods

The finite difference scheme with the shifted Grünwald formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, the boundary value method (BVM) was developed as a popular algorithm for solving the large systems of ODEs. This method requires the solutions of one or more nonsymmetric and large-scale linear systems. In this paper, the GMRES method with the block circulant preconditioner is proposed to solve relevant linear systems. Some conclusions about the convergence analysis and spectrum of the preconditioned matrices are also drawn if the diffusion coefficients are constant. Finally, extensive numerical experiments are reported to show the performance of our method for solving the fractional diffusion equations.