Unconditionally convergent numerical method for the fractional activator–inhibitor system with anomalous diffusion

Abstract In this paper, a novel numerical scheme is proposed to numerically solve the fractional activator–inhibitor system, which is a coupled two‐dimensional nonlinear model. In the temporal direction, we employ the Grünwald–Letnikov formula, in spatial direction, the Legendre spectral method is used. In terms of the error splitting argument technique, an optimal error estimate of the numerical scheme is obtained without any time‐step size conditions, while the usual analysis for high‐dimensional nonlinear fractional problems always required certain time‐step restrictions dependent on the spatial mesh size. Some numerical results are given to justify the theoretical analysis.