Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem

The supercooled Stefan problem and its variants describe the freezing of a liquid in physics, as well large system limits systemic risk models finance integrate-and-fire neuroscience. Adopting physics terminology, is known to feature finite-time blow-up rate for wide range initial temperature distributions liquid. Such can result discontinuity liquid-solid boundary. In this paper, we prove that natural Euler time-stepping scheme applied probabilistic formulation converges boundary physical solution globally time, Skorokhod M1 topology. course proof, give an explicit bound on local convergence scheme. We also run numerical tests compare our theoretical results practically observed behavior.