PCG method with Strang’s circulant preconditioner for Hermitian positive definite linear system in Riesz space fractional advection–dispersion equations

Abstract

In this paper, preconditioned conjugate gradient (PCG) method with Strang’s circulant preconditioner is investigated to solve the Hermitian positive definite linear systems, which is result from the Crank–Nicolson (C-N) finite difference scheme with the weighted and shifted Grunwald difference (WSGD) operators to discretize the Riesz space fractional advection–dispersion equation (RSFADE). We show that the spectrum of the preconditioned matrix is clustered around 1, and the singular values of the preconditioned matrix are uniformly bounded away from zero under a certain condition, respectively; hence the PCG method, when applied to solving the preconditioned system, converges superlinearly. Moreover, the complexity in each iteration of the PCG method is $$O(N\log N)$$ via using the fast Fourier transforms, where N is the matrix size. Numerical experiments are included to demonstrate the effectiveness of our approach.