The mean curvature flow describes the evolution of a surface (a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical method for mean curvature flows by using the Onsager principle as an approximation tool. We first show that the mean curvature flow can be derived naturally from the Onsager variational principle. Then we consider a piecewise linear approximation of the curve and derive a discrete geometric flow. The discrete flow is described by a system of ordinary differential equations for the nodes of the discrete curve. We prove that the discrete system preserve the energy dissipation structure in the framework of the Onsager principle and this implies the energy decreasing property. The ODE system can be solved by the improved Euler scheme and this leads to an efficient fully discrete scheme. We first consider the method for a simple mean curvature flow and then extend it to the volume preserving mean curvature flow and also a wetting problem on substrates. Numerical examples show that the method has optimal convergence rate and works well for all the three problems.

翻译：平均曲率流描述了曲面（或曲线）沿法向速度与局部平均曲率成正比的演化过程，在数学、科学与工程领域具有广泛应用。本文利用Onsager原理作为近似工具，为平均曲率流开发了一种数值方法。我们首先证明平均曲率流可以从Onsager变分原理自然导出。随后，采用分段线性近似对曲线进行离散，并推导出离散几何流。该离散流由离散曲线节点的一组常微分方程描述。我们证明该离散系统在Onsager原理框架下保持了能量耗散结构，这意味着能量递减性质成立。该常微分方程组可通过改进的欧拉格式求解，从而得到高效的全离散格式。我们首先将方法应用于简单平均曲率流，随后将其推广至体积保持平均曲率流以及基底上的润湿问题。数值算例表明该方法具有最优收敛速率，并在三类问题上均表现良好。