Block preconditioning strategies for time-space fractional diffusion equations

We present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time-space fractional diffusion equations. In contrast to the traditional time-marching procedure, the discretization via finite difference is considered in a fully coupled time-space framework. The resulting fully coupled discretized linear system is a summation of two Kronecker products. The four preconditioning methods are based on block diagonal, banded block triangular and Kronecker product splittings of the coefficient matrix. All preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator. Numerical experiments show the efficiency of the four block preconditioners, and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one.