We consider the Cahn-Hilliard equation with standard double-well potential. We employ a prototypical class of first order in time semi-implicit methods with implicit treatment of the linear dissipation term and explicit extrapolation of the nonlinear term. When the dissipation coefficient is held small, a conventional wisdom is to add a judiciously chosen stabilization term in order to afford relatively large time stepping and speed up the simulation. In practical numerical implementations it has been long observed that the resulting system exhibits remarkable stability properties in the regime where the stabilization parameter is $\mathcal O(1)$, the dissipation coefficient is vanishingly small and the size of the time step is moderately large. In this work we develop a new stability theory to address this perplexing phenomenon.

翻译：我们考虑Cahn-Hilliard方程式,该方程式具有标准的双well 潜能。 我们使用一种半隐含方法, 隐含地处理线性消散术语和非线性术语的外推法, 使用一种原型第一顺序的半隐含方法。 当消散系数保持较小时, 一种传统智慧是增加一个明智选择的稳定期, 以提供相对较大的时间加速和加速模拟。 在实际的数值实施中, 人们长期观察到, 由此形成的系统在稳定参数为$\mathcal O(1)$的制度中表现出显著的稳定性, 消散系数正在消失, 时间步骤的大小是中等的。 在这项工作中, 我们开发了一个新的稳定理论, 以解决这一令人困惑的现象。