We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

翻译：我们针对Cahn-Hilliard方程提出了有限体积格式，该格式无条件且离散地保持相场的有界性与自由能的耗散性。本文数值框架适用于多种自由能势函数（包括Ginzburg-Landau和Flory-Huggins势）、一般润湿边界条件及退化迁移率。其核心思想是迎风方法，我们将其与基于经典凸分裂方法的自由能项半隐式公式相结合。得益于该格式的维度分裂特性，其可自然扩展至任意维数，从而通过简单的并行化高效求解高维问题。通过不同维度、多种液滴-基底接触角的大量数值算例，验证并检验了所提数值格式的有效性。