On the Numerical Performance of Extrapolation‐Projection Linear L1 Scheme for Time‐Fractional Allen‐Cahn Dynamics

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This paper presents a simple linear, maximum bound principle‐preserving L1 scheme for the time‐fractional Allen‐Cahn (TFAC) equation. The scheme is built upon a novel extrapolation‐projection strategy that linearizes the nonlinear term and leads to a linear system with constant coefficients at each time step, substantially reducing computational cost and providing strong long‐time robustness. Although the extrapolation‐projection strategy does not alter the asymptotic convergence order inherent to the L1 approximation, the results indicate improved long‐time behavior through a more effective treatment of the nonlinear term. Moreover, the simulations exhibit a robust monotonic decay of a discrete energy functional. Comparative results show a clear improvement in numerical resolution over standard backward linearization approaches. Furthermore, the extrapolation‐projection framework is sufficiently general and can be readily extended to generalized TFAC equations with nonlinear free energy densities, including higher‐order polynomial and logarithmic potentials.