In this paper we propose and analyze a finite difference numerical scheme for the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition. The singular logarithmic potential is included in the Flory-Huggins energy expansion. Meanwhile, a dynamical evolution equation for the boundary profile corresponds to a lower-dimensional singular energy potential. In turn, a theoretical analysis for the coupled system becomes very challenging, since it contains nonlinear and singular energy potentials for both the interior region and on the boundary. In the numerical design, a convex splitting approach is applied to the chemical potential associated with the energy both at the interior region and on the boundary: implicit treatments for the singular and logarithmic terms, as well as the surface diffusion terms, combined with an explicit treatment for the concave expansive term. In addition, the discrete boundary condition for the phase variable is coupled with the evolutionary equation of the boundary profile. The resulting numerical system turns out to be highly nonlinear, singular and coupled. A careful finite difference approximation and convexity analysis reveals that such a numerical system could be represented as a minimization of a discrete numerical energy functional, which contains both the interior and boundary integrals. More importantly, all the singular terms correspond to a discrete convex functional. As a result, a unique solvability and positivity-preserving analysis could be theoretically justified, based on the subtle fact that the singular nature of the logarithmic terms around the singular limit values prevent the numerical solutions reaching these values. The total energy stability analysis could be established by a careful estimate over the finite difference inner product. Some numerical results are presented in this article.

翻译：本文针对具有动态边界条件的Flory-Huggins-Cahn-Hilliard方程，提出并分析了一种有限差分数值格式。Flory-Huggins能量展开中包含了奇异的对数势。同时，边界轮廓的动态演化方程对应于一个低维奇异能量势。因此，该耦合系统的理论分析变得极具挑战性，因为它同时包含了内部区域和边界上的非线性奇异能量势。在数值设计中，对内部区域和边界上与能量相关的化学势采用了凸分裂方法：对奇异项、对数项以及表面扩散项进行隐式处理，并结合对凹扩张项的显式处理。此外，相变量的离散边界条件与边界轮廓的演化方程相耦合。由此得到的数值系统具有高度非线性、奇异性和耦合性。通过细致的有限差分逼近和凸性分析，我们发现该数值系统可表示为离散数值能量泛函的最小化问题，该泛函同时包含内部积分和边界积分。更重要的是，所有奇异项对应于一个离散凸泛函。因此，基于对数项在奇异极限值附近的奇异性质能阻止数值解达到这些值的微妙事实，唯一可解性和保正性分析在理论上得以证明。总能量稳定性分析可通过有限差分内积的精细估计来建立。本文还展示了一些数值结果。