Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nurnberg (J. Comput. Phys., 222 (2007), pp. 441{467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which incorporates a mesh regularization technique when necessary, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as a mesh regularization technique when necessary, our proposed second-order scheme exhibits good properties with respect to the mesh distribution.

翻译：在过去的二十年中，曲线几何演化问题在科学计算领域引起了广泛关注。求解几何流最流行的数值方法之一是所谓的BGN格式，由Barrett、Garcke和Nurnberg提出（J. Comput. Phys., 222 (2007), pp. 441-467），因其计算效率和良好的网格性质等优越特性而受到青睐。然而，BGN格式的时间精度仅为一阶，开发更高阶的数值格式仍具挑战性。本文针对曲线几何流问题，提出一种全离散、时间二阶的参数化有限元方法，并在必要时引入网格正则化技术。该格式基于BGN公式、半隐式Crank-Nicolson蛙跳时间步进离散以及空间线性有限元近似构建。更重要的是，我们指出应使用流形距离和Hausdorff距离等形状度量（而非函数范数）来评估数值误差。大量数值实验表明，所提出的基于BGN的格式在形状度量下具有时间二阶精度。此外，通过必要时将经典BGN格式作为网格正则化技术，该二阶格式在网格分布方面展现出良好性质。