Fast TTTS iteration methods for implicit Runge-Kutta temporal discretization of Riesz space fractional advection-diffusion equations.

In this paper, we consider fast numerical methods for linear systems arising from implicit Runge-Kutta temporal discretization methods (based on the fourth-order, 2-stage Gauss method) for one- and two-dimensional Riesz space fractional advection-diffusion equations (RSFADEs). An implicit Runge-Kutta-standard/shifted Grünwald difference scheme for RSFADEs is introduced, and its stability and convergence are also studied. In the one-dimensional case, the coefficient matrix of the discretized linear system is the sum of an identity matrix, a Toeplitz matrix and a square of Toeplitz matrix. We construct a class of Toeplitz times Toeplitz splitting (TTTS) iteration methods to solve the corresponding linear systems. We prove that it converges uniformly to the exact solution without imposing any additional condition, and the optimal parameters for the TTTS iteration method are given. Meanwhile, we design an induced sine transform based preconditioner for two-dimensional problems to accelerate the convergence rate of the conjugate gradient method. Theoretically, we prove that the spectra of the preconditioned matrices of the proposed methods are clustering around 1. Numerical results are presented to illustrate the effectiveness of the proposed methods.