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In this paper, we develop an energy dissipative numerical scheme for gradientflows of planar curves, such as the curvature flow and the elastic flow. Ourstudy presents a general framework for solving such equations. To discretizetime, we use a similar approach to the discrete partial derivative method,which is a structure-preserving method for the gradient flows of graphs. Forthe approximation of curves, we use B-spline curves. Owing to the smoothness ofB-spline functions, we can directly address higher order derivatives. In thelast part of the paper, we consider some numerical examples of the elasticflow, which exhibit topology-changing solutions and more complicated evolution.Videos illustrating our method are available on YouTube.

Energy dissipative numerical scheme for gradient flows of planar curves using discrete partial derivatives and B-spline curves