In this paper, we present a new computational framework using coupled and decoupled approximations for a Cahn-Hilliard-Navier-Stokes model with variable densities and degenerate mobility. In this sense, the coupled approximation is shown to conserve the mass of the fluid, preserve the point-wise bounds of the density and decrease an energy functional. In contrast, the decoupled scheme is presented as a more computationally efficient alternative but the discrete energy-decreasing property can not be assured. Both schemes are based on a finite element approximation for the Navier-Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn-Hilliard part. Finally, several numerical experiments contrasting both approaches are conducted. In particular, results for a convergence test, a simple qualitative comparison and some well-known benchmark problems are shown.

翻译：本文针对具有可变密度和退化迁移率的Cahn-Hilliard-Navier-Stokes模型，提出了一种基于耦合与解耦逼近的新型计算框架。其中，耦合逼近能够保持流体质量守恒、密度逐点有界性并降低能量泛函；相比之下，解耦方案在计算效率上更具优势，但无法保证离散能量递减性质。两种方案均采用有限元逼近处理具有间断压力的Navier-Stokes流体流动，并使用间断Galerkin迎风格式处理Cahn-Hilliard部分。最后，通过数值实验对两种方法进行对比，展示了收敛性测试、简单定性比较及若干经典基准问题的计算结果。