Crank-Nicolson and ADI Finite Element Approaches for the Space-Fractional Gray-Scott Model

Abstract

This paper presents a novel and efficient numerical framework for solving the two-dimensional space-fractional Gray-Scott model. The proposed method combines the Crank-Nicolson scheme for temporal discretization with an Alternating Direction Implicit (ADI) Finite Element Method (FEM) for spatial approximation. A key innovation is the introduction of a stabilizing perturbation term, which facilitates the ADI splitting while preserving the method’s accuracy. Rigorous theoretical analysis proves that the fully-discrete scheme is unconditionally stable and achieves second-order convergence in time. Comprehensive numerical experiments are conducted to validate the theoretical findings, demonstrate the superior computational efficiency of the proposed ADI-FEM compared to existing methods, and explore the complex pattern formation dynamics governed by the fractional-order derivatives and model parameters.

Keywords

Gary-Scott model / Fractional calculus / Stability and convergence analysis / Pattern formation / 65M12 / 65N12 / 65N30 / 65N40

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