We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [{\it Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes}, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.

中文翻译：

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用于求解 Cahn-Hilliard Navier-Stokes 方程的全耦合框架：基于自适应八叉树网格的二阶能量稳定数值方法

我们提出了一个完全耦合的隐式时间框架，用于求解模拟两相流的热力学一致 Cahn-Hilliard Navier-Stokes 系统。在这项工作中，我们扩展了 Khanwale 等人提出的块迭代方法。[{\it 在基于并行自适应八叉树的网格上使用热力学一致能量稳定 Cahn-Hilliard Navier-Stokes 方程模拟两相流}，J. Comput。物理。(2020)]，到一个完全耦合的、可证明的二阶精确方案，同时保持能量稳定性。新方法在每次 Newton 迭代中需要更少的矩阵组件，从而缩短求解时间。该方法基于完全隐式的 Crank-Nicolson 时间方案和等阶 Galerkin 公式的压力稳定。那是，我们在空间中使用符合连续伽辽金 (cG) 有限元方法，并配备基于残差的变分多尺度 (RBVMS) 程序来稳定压力。我们使用基于并行八叉树的自适应网格将这种方法部署在大规模并行数值实现上。我们提出了全面的数值实验，显示了与经典案例文献结果的详细比较，包括单气泡上升、瑞利-泰勒不稳定性和盖子驱动的腔体流动问题。我们详细分析了我们的数值实现的缩放。我们提出了全面的数值实验，显示了与经典案例文献结果的详细比较，包括单气泡上升、瑞利-泰勒不稳定性和盖子驱动的腔体流动问题。我们详细分析了我们的数值实现的缩放。我们提出了全面的数值实验，显示了与经典案例文献结果的详细比较，包括单气泡上升、瑞利-泰勒不稳定性和盖子驱动的腔体流动问题。我们详细分析了我们的数值实现的缩放。