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:我们为Cahn-Hilliard等式提出了数量有限的计划,无条件和独立地保持了阶段场的界限和自由能源的分散。我们的数字框架适用于各种自由能源潜力,包括Ginzburg-Landau和Flory-Huggins,一般的湿边界条件,以及堕落的动员。它的中心是上风方法,我们结合了基于经典的锥形分裂法的对自由能源条件的半暗配方。计划扩大至任意的多个维度是直接的,因为它们的分层性质使得能够以简单的平行方式有效地解决更高维度的问题。数字计划通过不同层面的各种实例以及滴子和子体之间的不同接触角度得到验证和测试。