We propose a novel formulation for parametric finite element methods to simulate surface diffusion of closed curves, which is also called as the curve diffusion. Several high-order temporal discretizations are proposed based on this new formulation. To ensure that the numerical methods preserve geometric structures of curve diffusion (i.e., the perimeter-decreasing and area-preserving properties), our formulation incorporates two scalar Lagrange multipliers and two evolution equations involving the perimeter and area, respectively. By discretizing the spatial variable using piecewise linear finite elements and the temporal variable using either the Crank-Nicolson method or the backward differentiation formulae method, we develop high-order temporal schemes that effectively preserve the structure at a fully discrete level. These new schemes are implicit and can be efficiently solved using Newton's method. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, as measured by the manifold distance, while simultaneously preserving the geometric structure of the curve diffusion.

翻译：我们提出了一种新颖的参数化有限元方法公式，用于模拟封闭曲线的表面扩散，也称为曲线扩散。基于这一新公式，提出了几种高阶时间离散方案。为确保数值方法保持曲线扩散的几何结构（即周长递减和面积守恒特性），我们的公式引入了两个标量拉格朗日乘子，并分别建立了涉及周长和面积的两个演化方程。通过采用分段线性有限元对空间变量进行离散，并运用Crank-Nicolson方法或向后微分公式方法对时间变量进行离散，我们构建了在完全离散层面上有效保持结构的高阶时间格式。这些新格式是隐式的，可通过牛顿法高效求解。大量数值实验表明，我们的方法在通过流形距离衡量的时间精度方面达到了预期目标，同时保持了曲线扩散的几何结构。