  The supercooled Stefan problem and its variants describe the freezing of a
supercooled liquid in physics, as well as the large system limits of systemic
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
discontinuity of the liquid-solid boundary. In this paper, we prove that the
natural Euler time-stepping scheme applied to a probabilistic formulation of
the supercooled Stefan problem converges to the liquid-solid boundary of its
physical 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.
