We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions. We employ the standard semi-implicit numerical scheme which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Under natural constraints on the time step we prove strict phase separation and energy stability of the semi-implicit scheme. This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.
翻译:我们认为,二维卡恩-希利亚德方程式具有对数潜力和周期边界条件。我们采用了标准的半隐含数字办法,以隐含的方式处理线性第四级解体术语和非线性术语。在对时间步骤的自然限制下,我们证明半隐含性计划的严格阶段分离和能源稳定性。这似乎是Cahn-希利亚德方程式半隐性分离并具有独特潜力的第一个严格结果。