In this paper, we construct a new class of predictor–corrector time-stepping schemes for the Cahn–Hilliard equation, which are linear, second-order accurate in time, unconditionally energy stable, and uniquely solvable. Then, we present the stability and error estimates of the semi-discrete numerical schemes for solving the Cahn–Hilliard equation with general nonlinear bulk potentials. The semi-discrete scheme is further discretized using the compact central finite difference method. Several numerical examples are shown to verify the theoretical results. In particular, the numerical simulations show that the predictor–corrector schemes reach the second-order convergence rate at relatively larger time-step sizes than the classical linear schemes. The numerical strategies and theoretical tools developed in this article could be readily applied to study other phase-field models or models that can be cast as gradient flow problems.

在本文中，我们为Cahn-Hilliard方程构造了一类新的预测器-校正器时间步长方案，该方案是线性的，时间上的二阶精确度，无条件的能量稳定且可唯一求解。然后，我们提出了半离散数值格式的稳定性和误差估计，用于求解具有一般非线性体势的Cahn-Hilliard方程。使用离散中心有限差分法进一步离散半离散方案。显示了几个数值示例，以验证理论结果。特别是，数值模拟表明，与经典线性方案相比，预测器-校正器方案在相对较大的时间步长上达到了二阶收敛速度。