An efficient Crank–Nicolson scheme with preservation of the maximum bound principle for the high-dimensional Allen–Cahn equation

Abstract

In this study, we present a linear second-order single time-stepping finite difference scheme for solving the Allen–Cahn equation. The temporal integration is realized by combining the predictor-correction fashion of the Crank–Nicolson scheme with a linear stabilization technique, where central finite differences are employed for spatial discretization. In contrast to the BDF2 scheme, the proposed method operates without any extrapolation strategies, avoiding the need to compute the ratio of adjacent time steps during each time iteration. The discrete Maximum bound principle (MBP) is proven under the mild constraints on the time step size. The convergence analysis in $L^2$ and $L^{\infty }$ norms is also presented as well as the energy stability. Several typical 2D and 3D numerical experiments are carried out to verify the theoretical results and demonstrate the efficiency of the proposed scheme.