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