A linear second-order finite difference scheme for the Allen-Cahn equation with a general mobility

In this paper, a linear second-order finite difference scheme is proposed for the Allen-Cahn equation with a general positive mobility. The Crank-Nicolson scheme and Taylor's formula are used for temporal discretization, and the central finite difference method is used for spatial approximation. The discrete maximum bound principle (MBP), the discrete energy stability and $L^\infty$-norm error estimation are discussed, respectively. Finally, some numerical examples are presented to verify our theoretical results.