We present a novel spatial discretization for the Cahn-Hilliard equation including transport. The method is given by a mixed discretization for the two elliptic operators, with the phase field and chemical potential discretized in discontinuous Galerkin spaces, and two auxiliary flux variables discretized in a divergence-conforming space. This allows for the use of an upwind-stabilized discretization for the transport term, while still ensuring a consistent treatment of structural properties including mass conservation and energy dissipation. Further, we couple the novel spatial discretization to an adaptive time stepping method in view of the Cahn-Hilliard equation's distinct slow and fast time scale dynamics. The resulting implicit stages are solved with a robust preconditioning strategy, which is derived for our novel spatial discretization based on an existing one for continuous Galerkin based discretizations. Our overall scheme's accuracy, robustness, efficient time adaptivity as well as structure preservation and stability with respect to advection dominated scenarios are demonstrated in a series of numerical tests.

翻译：本文提出了一种用于含输运项的Cahn-Hilliard方程的新型空间离散方法。该方法通过对两个椭圆算子进行混合离散实现：相场与化学势采用间断Galerkin空间离散，两个辅助通量变量采用散度协调空间离散。这种设计允许对输运项采用迎风稳定化离散，同时仍能确保结构性质（包括质量守恒与能量耗散）的一致性处理。此外，针对Cahn-Hilliard方程特有的慢速与快速时间尺度动力学特性，我们将新型空间离散方法与自适应时间步进方法相结合。所得隐式求解阶段采用鲁棒预处理策略进行求解，该策略基于现有连续Galerkin离散的预处理方法，并针对我们提出的新型空间离散形式进行了推导。通过一系列数值实验，验证了本整体格式在精度、鲁棒性、高效时间自适应性以及结构保持特性方面的优越性，并证明了其在平流主导场景下的稳定性。