A Numerical Method for Tracking Curve Networks Moving with Curvature Motion

A finite difference method is proposed to track curves whose normal velocity is given by their curvature and which meet at different types of junctions. The prototypical example is that of phase interfaces that meet at prescribed angles, although eutectic junctions and interactions through nonlocal effects are also considered. The method is based on a direct discretization of the underlying parabolic problem and boundary conditions. A linear stability analysis is presented for our scheme as well as computational studies that confirm the second order convergence to smooth solutions. After a singularity in the curve network where the solution is no longer smooth, we demonstrate "almost" second-order convergence. A numerical study of singularity types is done for the case of networks that meet at prescribed angles at triple junctions. Finally, different discretizations and methods for implicit time stepping are presented and compared.