A Fourth-Order Compact ADI Scheme for Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation

TL;DR

This paper introduces a high-order compact ADI scheme for efficiently solving two-dimensional Riesz space fractional nonlinear reaction-diffusion equations, achieving high accuracy and stability.

Contribution

It develops a novel fourth-order compact ADI scheme combined with BDF2 for time, providing improved accuracy and stability for fractional PDEs.

Findings

The scheme is stable and convergent.

Numerical experiments confirm high accuracy.

Method outperforms lower-order schemes.

Abstract

In this paper, a second-order backward difference formula (abbr. BDF2) is used to approximate first-order time partial derivative, the Riesz fractional derivatives are approximated by fourth-order compact operators, a class of new alternating-direction implicit difference scheme (abbr. ADI) is constructed for two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. Stability and convergence of the numerical method are analyzed. Numerical experiments demonstrate that the proposed method is effective.