A novel fourth-order scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations and its optimal preconditioned solver

TL;DR

This paper introduces a high-order finite difference scheme for 2D Riesz space fractional reaction-diffusion equations, proving its stability and convergence, and develops an optimal preconditioned iterative solver with mesh-independent convergence.

Contribution

It presents the first preconditioned iterative solver with mesh-independent convergence for a high-order scheme solving 2D Riesz fractional equations.

Findings

The scheme is unconditionally stable and convergent.

The preconditioner accelerates convergence with a mesh-independent rate.

Numerical results confirm the scheme's accuracy and solver efficiency.

Abstract

A novel fourth-order finite difference formula coupling the Crank-Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction-diffusion equations in two dimensions. Theoretically, under the Lipschitz assumption on the nonlinear term, the proposed high-order scheme is proved to be unconditionally stable and convergent in the discrete $L_2$-norm. Moreover, a $\tau$-matrix based preconditioner is developed to speed up the convergence of the conjugate gradient method with an optimal convergence rate (a convergence rate independent of mesh sizes) for solving the symmetric discrete linear system. Theoretical analysis shows that the spectra of the preconditioned matrices are uniformly bounded in the open interval $(3/8,2)$. To the best of our knowledge, this is the first attempt to develop a preconditioned iterative solver with a mesh-independent convergence rate for the linearized high-order scheme. Numerical examples are given to validate the accuracy of the scheme and the effectiveness of the proposed preconditioned solver.