Convergence of the Original BGN Method for Mean Curvature Flow

This paper provides a rigorous convergence proof for the original Barrett-Garcke-Nürnberg (BGN) method applied to mean curvature flow of curves. Using backward error analysis techniques borrowed from ODE numerical analysis, the authors show that the method converges in the manifold distance with order h² when the time step τ equals h². The key innovation lies in constructing an approximate flow by combining discrete velocity expansions in space and time.