A linear second-order maximum bound principle preserving finite difference scheme for the generalized Allen–Cahn equation

In this paper, we consider the numerical method for generalized Allen–Cahn equation with nonlinear mobility and convection term. We propose a linear second-order finite difference scheme which preserves the discrete maximum bound principle (MBP). The scheme is discretized by stabilized Crank–Nicolson in time, upwind scheme for convection term and central-difference scheme for diffusion term. We show that the proposed scheme preserves the discrete MBP under some constraints on temporal step size and stabilizing parameter. Optimal $L^{\infty }$-error estimate is obtained for our scheme. Several numerical experiments are performed to validate our theoretical results.