A filtered time stepping scheme for curve shortening flow for open and closed curves

We propose a filtered time stepping finite element scheme for curve shortening flow of open and closed curves in arbitrary codimension that is second-order accurate in time. Open curves are assumed to evolve inside a given domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, and meet the external boundary $\mathrm{\partial }\mathrm{\Omega }$ orthogonally. We prove optimal error bounds for the $L^2$– and $H^1$–norms. In practice only a single linear system needs to be solved at each time step. Numerical experiments confirm the accuracy and practicality of the introduced method, including an asymptotic equidistribution property.