# A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations

## TL;DR

This paper introduces a second-order stabilized semi-implicit Fourier spectral method for solving nonlinear space-fractional reaction-diffusion equations, ensuring stability without CFL constraints and providing optimal error estimates.

## Contribution

The work develops a novel stabilized semi-implicit spectral method for fractional reaction-diffusion equations with proven stability and error bounds, applicable to complex biological and physical models.

## Key findings

- Method is unconditionally stable and second-order accurate.
- Successfully applied to fractional Allen-Cahn, Gray-Scott, and FitzHugh-Nagumo models.
- Revealed distinct pattern formation behaviors in fractional models.

## Abstract

The reaction-diffusion model can generate a wide variety of spatial patterns, which has been widely applied in chemistry, biology, and physics, even used to explain self-regulated pattern formation in the developing animal embryo. In this work, a second-order stabilized semi-implicit time-stepping Fourier spectral method is presented for the reaction-diffusion systems of equations with space described by the fractional Laplacian. We adopt the temporal-spatial error splitting argument to illustrate that the proposed method is stable without imposing the CFL condition, and we prove an optimal L2-error estimate. We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical criterion to choose the time step size to guarantee the stability of the semi-implicit method. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator, which are quite different from the patterns of the corresponding integer-order model.