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In this paper, we propose a linear second-order numerical method for solving the Allen-Cahn equation with general mobility. The fully-discrete scheme is achieved by using the Crank-Nicolson formula for temporal integration and the central difference method for spatial approximation, together with two additional stabilization terms. Under mild constraints on the two stabilizing parameters, the proposed numerical scheme is shown to unconditionally preserve the discrete maximum bound principle and the discrete original energy dissipation law. Error estimate in the 𝐿∞ norm is successfully derived for the proposed scheme. Finally, some numerical experiments are conducted to verify the theoretical results and demonstrate the performance of the proposed scheme in combination with an adaptive time-stepping strategy. 

In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete $H^1$error estimate and energy stability for the classic constant mobility case and the $L^{\mathrm{\infty <\#comment/>}}$error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy.