We propose and analyze a linearly stabilized semi-implicit diffusive Crank–Nicolson scheme for the Cahn–Hilliard gradient flow. In this scheme, the nonlinear bulk force is treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation. Rigorous error analysis is carried out for the fully discrete scheme. When the time step-size and the space step-size are small enough, second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of $1∕\epsilon$. Here $\epsilon$ is the thickness of the interface. Numerical results together with an adaptive time stepping are presented to verify the accuracy and efficiency of the proposed scheme.

我们提出并分析了Cahn-Hilliard梯度流的线性稳定半隐式扩散Crank-Nicolson方案。在该方案中，非线性本体力用两个二阶稳定项明确处理。这种处理导致具有恒定系数和可证明的离散能量耗散的线性椭圆系统。对完全离散的方案进行严格的误差分析。当时间步长和空间步长足够小时，可以通过由某个较低阶多项式控制的预因子来获得时间的二阶精度。$\mathrm{1个}∕\epsilon$。这里$\epsilon$是界面的厚度。数值结果与自适应时间步长一起被提出来验证所提方案的准确性和效率。