Fast and accurate high-order method for high dimensional space-fractional reaction-diffusion equation with general boundary conditions

TL;DR

This paper introduces a fast, stable, and highly accurate explicit numerical method for high-dimensional space-fractional reaction-diffusion equations, combining high-order spatial discretization with efficient time integration.

Contribution

It develops a novel high-order explicit scheme using compact finite differences, matrix transfer technique, and exponential time differencing for fractional reaction-diffusion equations.

Findings

Achieves high accuracy and stability in 2D and 3D models

Demonstrates efficiency through FFT-based implementation

Validates effectiveness on multiple reaction-diffusion systems

Abstract

To achieve efficient and accurate long-time integration, we propose a fast, accurate, and stable high-order numerical method for solving fractional-in-space reaction-diffusion equations. The proposed method is explicit in nature and utilizes the fourth-order compact finite difference scheme and matrix transfer technique (MTT) in space with FFT-based implementation. Time integration is done through the accurate fourth-order modified exponential time differencing Runge-Kutta scheme. The linear stability analysis and various numerical experiments including two-dimensional (2D) Fitzhugh-Nagumo, Gierer-Meinhardt, Gray-Scott and three-dimensional (3D) Schnakenberg models are presented to demonstrate the accuracy, efficiency, and stability of the proposed method.