We present a high–order discontinuous Galerkin (DG) discretization for the three–phase Cahn–Hilliard model of [Boyer, F., & Lapuerta, C. (2006). Study of a three component Cahn–Hilliard flow model]. In this model, consistency is ensured with an additional term in the chemical free–energy. The model considered in this work includes a wall boundary condition that allows for an arbitrary equilibrium contact angle in three–phase flows. The model is discretized with a high–order discontinuous Galerkin spectral element method that uses the symmetric interior penalty to compute the interface fluxes, and allows for unstructured meshes with curvilinear hexahedral elements. The integration in time uses a first order IMplicit–EXplicit (IMEX) method, such that the associated linear systems are decoupled for the two Cahn–Hilliard equations. Additionally, the Jacobian matrix is constant, and identical for both equations. This allows us to solve the two systems by performing only one LU factorization, with the size of the two–phase system, followed by two Gauss substitutions. Finally, we test numerically the accuracy of the scheme providing convergence analyses for two and three–dimensional cases, including the captive bubble test, the study of two bubbles in contact with a wall and the spinodal decomposition in a cube and in a curved pipe with a “T” junction.

我们为[Boyer，F.和Lapuerta，C.（2006）的三相Cahn-Hilliard模型提供了一个高阶不连续Galerkin（DG）离散化。三分量Cahn–Hilliard流动模型的研究]。在此模型中，化学自由能中的另一个术语确保了一致性。在这项工作中考虑的模型包括一个壁边界条件，该条件允许在三相流中具有任意的平衡接触角。该模型使用高阶不连续Galerkin光谱元素方法离散化，该方法使用对称内部罚分法计算界面通量，并允许使用具有曲线六面体元素的非结构化网格。时间上的积分使用一阶隐式-显式（IMEX）方法，这样对于两个Cahn-Hilliard方程，相关联的线性系统就解耦了。此外，雅可比矩阵是常数，并且对于两个方程都是相同的。这使我们能够通过仅执行一次LU分解来解决这两个系统，并采用两相系统的大小，然后进行两次高斯替换。最后，我们通过数值测试该方案的准确性，为二维和三维情况提供收敛性分析，包括俘获气泡测试，研究与壁接触的两个气泡以及立方体和弯曲管中的旋节线分解。一个“ T”形结。