In this paper, we present a fourth-order difference scheme for solving the Allen-Cahn equation in both 1D and 2D. The proposed scheme is described by the compact difference operators together with the additional stabilized term. As a matter of fact, the Allen-Cahn equation contains the nonlinear reaction term which is eminently proved that numerical schemes are mostly nonlinear. To solve the complexity of nonlinearity, the Crank-Nicolson/Adams-Bashforth method is applied in order to deal with the nonlinear terms with the linear implicit scheme. The well-known energy-decaying property of the equation is maintained by the proposed scheme in the discrete sense. Additionally, the $L_{\infty }$ error analysis is carried out in the 1D case in a rigorous way to show that the method is fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. Concurrently, we examine the $L_2$ and $H^1$ error analysis for the scheme in the case of 2D. We consider the impact of the additional stabilized term on numerical solutions. The consequences confirm that an appropriate value of the stabilized term yields a significant improvement. Moreover, relevant results are carried out in the numerical simulations to illustrate the faithfulness of the present method by the confirmation of existing pieces of evidence.

在本文中，我们提出了求解一维和二维艾伦卡恩方程的四阶差分格式。所提出的方案由紧差分算子和附加的稳定项一起描述。事实上，Allen-Cahn 方程包含非线性反应项，这已显着证明数值格式大多是非线性的。为了解决非线性的复杂性，应用了Crank-Nicolson/Adams-Bashforth方法，以便用线性隐式格式处理非线性项。所提出的方案在离散意义上保持了该方程众所周知的能量衰减特性。此外，${\mathrm{大号}}_{\infty }$在一维情况下以严格的方式进行误差分析，以表明该方法在空间和时间步长上分别具有四阶和二阶精度。同时，我们考察${\mathrm{大号}}_2$和$H^1$方案在二维情况下的误差分析。我们考虑附加稳定项对数值解的影响。结果证实，适当的稳定项值会产生显着的改善。此外，在数值模拟中进行了相关结果，通过对现有证据的确认来说明本方法的真实性。