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To achieve efficient and accurate long-time integration, we propose a fast,accurate, and stable high-order numerical method for solvingfractional-in-space reaction-diffusion equations. The proposed method isexplicit in nature and utilizes the fourth-order compact finite differencescheme and matrix transfer technique (MTT) in space with FFT-basedimplementation. Time integration is done through the accurate fourth-ordermodified exponential time differencing Runge-Kutta scheme. The linear stabilityanalysis 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, andstability of the proposed method.

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