变密度Cahn--Hilliard--Navier--Stokes流在演化网格上的历史相容能量稳定有限元格式
History-Compatible Energy-Stable Finite Element Schemes for Variable-Density Cahn--Hilliard--Navier--Stokes Flows on Evolving Meshes
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- School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
- National Center for Applied Mathematics in Hunan, Xiangtan University（湖南应用数学中心（湘潭大学））
- Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University（湘潭大学湖南省工程计算与仿真重点实验室）
- Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods, Xiangtan University（湘潭大学湖南省基础学科基本算法理论与新型计算方法研究中心）
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中文总结 AI 辅助
针对变密度CHNS流在演化网格上的离散问题，提出解耦的BE和BDF2格式，结合精确跨网格配对与历史表示，实现能量稳定、线性子问题及二阶收敛，实验验证了守恒与动力学。
中文摘要 AI 辅助
我们考虑在有限元网格上的变密度Cahn--Hilliard--Navier--Stokes (CHNS)离散格式，这些网格可在接受的时间层之间通过固定拓扑运动或拓扑改变的重网格而变化。当离散空间随时间变化时，多步格式中进入的相、动能和压力历史在不同的离散结构中测量，通常不能通过单一算子转移。我们通过将精确的物理跨网格配对与对应于相应相能、动能和压力梯度存储的历史表示相结合，开发了解耦的向后欧拉(BE)和二阶向后差分公式(BDF2)格式。相更新还确定了用于动量输运的Abels--Garcke--Grün一致的的质量通量。一个标量毛细交换方程将相和流体求解分开，同时保留离散能量交换。所得的场子问题是线性的，标量方程具有唯一的正解，并且在所述可接受性假设下，格式满足修正的能量平衡，无需时间步长限制。数值实验证实了在两种网格更新下的二阶时间收敛性、相质量守恒、无外力测试中的修正能量衰减，以及可比的Rayleigh--Taylor和上升气泡动力学。
英文摘要
We consider variable-density Cahn--Hilliard--Navier--Stokes (CHNS) discretizations on finite element meshes that may change between accepted time levels through fixed-topology motion or topology-changing remeshing. When the discrete spaces vary in time, the phase, kinetic, and pressure histories entering a multistep scheme are measured in different discrete structures and cannot, in general, be transferred by a single operator. We develop decoupled backward Euler (BE) and second-order backward differentiation formula (BDF2) schemes by combining exact physical cross-mesh pairings with history representations compatible with the corresponding phase-energy, kinetic-energy, and pressure-gradient storages. The phase update also determines an Abels--Garcke--Grün-consistent mass flux used in the momentum transport. A scalar capillary-exchange equation separates the phase and fluid solves while retaining the discrete energy exchange. The resulting field subproblems are linear, the scalar equation has a unique positive solution, and the schemes satisfy modified energy balances without a time-step restriction under the stated admissibility assumptions. Numerical experiments confirm second-order temporal convergence under both mesh updates, phase-mass conservation, modified-energy decay in the unforced tests, and comparable Rayleigh--Taylor and rising-bubble dynamics.
