发表机构
Hong Kong Polytechnic University; The University of Massachusetts, North Dartmouth; Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Beijing Normal University（香港理工大学; 马萨诸塞大学达特茅斯分校; 中国科学院数学与系统科学研究院; 北京师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带动力学边界条件的Cahn-Hilliard-Navier-Stokes系统，提出并分析了一种有限差分格式，通过凸分裂和半隐式离散化处理非线性，理论证明了最优阶收敛性和能量稳定性，并首次为移动接触线问题提供了收敛性理论证明。
AI 中文摘要
本文提出并分析了一种用于Cahn-Hilliard-Navier-Stokes系统的有限差分数值格式，该系统与动力学边界条件相结合。此类物理系统在移动接触线问题中具有潜在应用。边界轮廓由低维能量势控制，并与相变量的非齐次边界条件耦合。在数值设计中，对体相和表面层面的化学势采用凸分裂方法，这导致了一个高度耦合的非线性系统。在非线性流体对流以及流体运动与相变量演化之间的耦合项中采用了半隐式离散化。仔细的有限差分近似和凸性分析表明，该数值系统可以表示为与流体对流相关的非对称单调映射。进而，基于单调性论证，唯一可解性成立。通过仔细的求和-分部计算获得了总能量稳定性分析。特别地，本文从理论上建立了最优阶收敛性分析。精确解的离散质量守恒需要保持误差函数的零均值性质，以便相关的离散H_h^{-1}范数有定义。结合傅里叶投影和辅助函数来克服这一困难。此外，粗略和精细误差估计的方法得出了所需的收敛结果。本文呈现了一些数值结果，证明了所提数值格式的稳健性。据我们所知，这项工作首次在文献中为移动接触线问题的数值格式提供了收敛性分析和误差估计的理论证明。
英文摘要
A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Navier-Stokes system, combined with a dynamical boundary condition. Such a physical system has potential applications in the moving contact line problem. The boundary profile is governed by a lower-dimensional energy potential, coupled with a non-homogeneous boundary condition for the phase variable. In the numerical design, a convex-splitting approach is applied to the chemical potential in both bulk and surface levels, which leads to a highly coupled nonlinear system. A semi-implicit discretization is taken in the nonlinear fluid convection, as well as the coupled terms between the fluid motion and phase variable evolution. A careful finite difference approximation and convexity analysis reveals that such a numerical system could be represented as a non-symmetric and monotone mapping associated with the fluid convection. In turn, the unique solvability is valid based on the monotonicity argument. The total energy stability analysis is obtained through a careful summation-by-part calculation. In particular, an optimal rate convergence analysis is theoretically established in this work. The discrete mass conservation of the exact solution is required to preserve the mean-zero property of the error function, so that the associated discrete H_h^{-1} norm is well-defined. A combination of the Fourier projection and an auxiliary function is applied to overcome this difficulty. Furthermore, an approach of rough and refined error estimates concludes the desired convergence result. Some numerical results are presented in this article, which demonstrate the robustness of the proposed numerical scheme. In our knowledge, this work provides a theoretical proof of convergence analysis and error estimate for a numerical scheme to the moving contact line problem, for the first time in the literature.
