A preconditioner based on sine transform for two-dimensional Riesz space factional diffusion equations in convex domains

TL;DR

This paper introduces a fast numerical method utilizing a sine transform-based preconditioner to efficiently solve two-dimensional Riesz space fractional diffusion equations with nonlinear sources in convex domains, demonstrating stability and effectiveness.

Contribution

The paper develops a novel preconditioner based on the sine transform for solving discretized Riesz fractional diffusion equations efficiently.

Findings

Preconditioner accelerates convergence of iterative solvers.

Method is stable and convergent for convex domain problems.

Numerical experiments confirm improved computational efficiency.

Abstract

In this paper, we develop a fast numerical method for solving the time-dependent Riesz space fractional diffusion equations with a nonlinear source term in the convex domain. An implicit finite difference method is employed to discretize the Riesz space fractional diffusion equations with a penalty term in a rectangular region by the volume-penalization approach. The stability and the convergence of the proposed method are studied. As the coefficient matrix is with the Toeplitz-like structure, the generalized minimum residual method with a preconditioner based on the sine transform is exploited to solve the discretized linear system, where the preconditioner is constructed in view of the combination of two approximate inverse ${\tau}$ matrices, which can be diagonalized by the sine transform. The spectrum of the preconditioned matrix is also investigated. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.