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 N\"urnberg (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 integrates with two different mesh regularization techniques, 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 mesh regularization techniques, our proposed second-order schemes exhibit good properties with respect to the mesh distribution. In addition, an unconditional interlaced energy stability property is obtained for one of the mesh regularization techniques.

翻译：在过去的二十年里，几何曲线演化领域引起了科学计算界的广泛关注。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Nürnberg提出（J. Comput. Phys., 222 (2007), pp. 441–467），因其优越的特性（如计算效率和良好的网格性质）而备受青睐。然而，BGN格式在时间上仅限于一阶精度，如何发展更高阶的数值格式具有挑战性。本文提出了一种全离散、时间二阶精度的参数有限元方法，该方法结合了两种不同的网格正则化技术，用于求解曲线的几何流。该格式基于BGN公式、半隐式Crank-Nicolson蛙跳时间步进离散化以及空间上的线性有限元逼近构建而成。更重要的是，我们指出应采用形状度量（如流形距离和Hausdorff距离）而非函数范数来衡量数值误差。大量数值实验表明，所提出的基于BGN的格式在形状度量意义下具有时间二阶精度。此外，通过采用经典BGN格式作为网格正则化技术，我们提出的二阶格式在网格分布方面展现出良好特性。另外，针对其中一种网格正则化技术，我们获得了无条件交错的能量稳定性。