Numerical simulation of the three dimensional Allen–Cahn equation by the high-order compact ADI method

In this paper, a new linearized high-order compact difference method is presented for numerical simulation of three dimensional (3D) Allen–Cahn equation with three kinds of boundary conditions. The method, which is based on the Crank–Nicholson/Adams–Bashforth scheme combined with the Douglas–Gunn ADI method, is second order accurate in time and fourth order accurate in space and energy degradation. The main advantages of this method is that the nonlinear penalty term f(u)f(u) is linear and an extra stabilizing term is added to alleviate the stability constraint while maintaining accuracy and simplicity. Numerical experiments are given to demonstrate the validity and applicability of the new method.