  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.
