A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

Abstract

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.