发表机构
School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出轴对称平均曲率流的自适应移动网格方法，基于等分布原理和曲率监控函数，结合有限差分与能量稳定格式，有效提升复杂几何演化的精度并防止网格退化。
AI 中文摘要
本文针对轴对称平均曲率流的数值模拟，提出了自适应移动网格方法，涵盖各向同性和各向异性两种情况。这些方法在网格等分布原理的框架下发展而来，其中采用精心设计的切向速度，在演化过程中动态地重新分布网格点。为了准确捕捉演化界面的关键几何特征，我们基于曲率 $\kappa$、其弧长导数 $\kappa_s$ 以及曲率平方 $\kappa^2$ 来选取监控函数。这些监控函数可灵活调整以适应不同的问题设置，并在决定最终网格质量和数值精度方面起着至关重要的作用。空间离散采用中心有限差分，时间积分则使用一阶和二阶时间步进格式，包括 BDFk（$k=1,2$）和 Crank-Nicolson 方法。此外，在自适应系统中引入拉格朗日乘子方法以强制执行底层几何约束，从而得到能量稳定的数值格式。数值实验证实了所提方法的收敛性和能量稳定性。更重要的是，结果清楚地表明，所提方法在复杂几何演化中具有显著优势：通过利用适当设计的监控函数，自适应方法实现了网格点的动态重新分布，高效捕获局部几何特征，显著提高数值精度，并有效防止网格退化，特别是在各向异性情形下。
英文摘要
This paper introduces adaptive moving mesh methods for the numerical simulation of axisymmetric mean curvature flow, addressing both isotropic and anisotropic cases. The methods are developed within the framework of the mesh equidistribution principle, where a carefully designed tangential velocity is employed to dynamically redistribute mesh points during the evolution. To accurately capture the key geometric features of the evolving interfaces, we select monitor functions based on the curvature $κ$, its arc-length derivative $κ_s$, and the squared curvature $κ^2$. These monitor functions can be flexibly tailored to suit different problem settings and play a vital role in determining the resulting mesh quality and numerical accuracy. Spatial discretization is performed using central finite differences, while temporal integration is handled with first- and second-order time-stepping schemes, including the BDFk ($k=1,2$) and Crank-Nicolson methods. Additionally, a Lagrange multiplier approach is incorporated into the adaptive system to enforce the underlying geometric constraint, resulting in energy-stable numerical schemes. Numerical experiments confirm the convergence and energy stability of the proposed methods. More importantly, the results clearly show that the proposed methods offer significant advantages in complex geometric evolutions: by utilizing appropriately designed monitor functions, the adaptive methods achieve dynamic redistribution of mesh points, efficiently capturing localized geometric features, significantly improving numerical accuracy, and effectively preventing mesh degeneration, particularly in anisotropic cases.
Comments32 pages, 26 figures
