Preconditioning techniques and parameterized regular splitting iteration methods for the Riesz distributed-order space-fractional diffusion equations with variable coefficients

Abstract

In this paper, we consider fast algorithm and parameterized regular splitting iteration methods for solving the distributed-order Riesz space fractional diffusion equations (DO-RSFDEs) with variable diffusion coefficients. We use the mid-point quadrature and shifted Grünwald and implicit finite difference (MPQ-SG-IFD) scheme to discretize the considered equation. The MPQ-SG-IFD scheme is unconditionally stable and has a global truncation error of $O(\Delta {\alpha }^2+h+\tau )$. The preconditioned conjugate gradient (PCG) method with a symmetric and positive definite Toeplitz (SPDT) preconditioner is employed to solve the discretized linear system. The SPDT preconditioner can be viewed as a modification of the Toeplitz preconditioner proposed by Yang et al. (2023) [34]), but unlike the Toeplitz preconditioner, we prove that the spectrum of the preconditioned matrix is bounded above by positive constants independent of the temporal step size and the spatial grid size. In particular, the lower bound of the spectrum is equal to 1. For two-dimensional DO-RSFDEs, we present two classes of parameterized regular splitting (RS) iteration methods to solve the discretized linear system. The convergence of these two parameterized RS iterative methods is theoretically established. Numerical results are presented to demonstrate the effectiveness of our proposed methods.