A maximum-preserving high-order scheme with relaxed constraints for the space-fractional Allen–Cahn equation

In this paper, we investigate the numerical solution of the space-fractional Allen–Cahn equation, a fundamental model widely used to describe phase separation phenomena in material science and fluid dynamics. Our work begins by re-deriving a fourth-order numerical differentiation formula for approximating the Riesz derivative, developed from the perspective of generating functions. For time discretization, we employ the Crank–Nicolson method and introduce a novel technique to handle the nonlinear term effectively. By integrating these approaches, a new fully discrete finite difference scheme is constructed, achieving high-order accuracy in both space and time. One of the key contributions of this study is the design of a second-order approximation for the nonlinear term. This construction not only preserves structural properties but also enables the establishment of a discrete maximum principle and energy stability under less stringent conditions compared to the existing methods. Moreover, a rigorous theoretical analysis is provided, confirming that the proposed scheme attains the expected convergence order of Oτ2+h4. Finally, the theoretical findings are validated through a series of comprehensive numerical experiments. The results robustly demonstrate the method’s practical effectiveness, corroborating its high accuracy and computational efficiency.