  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 $L^{\infty}$
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.
