2020/10/11 by Vadim Kaushansky, Christoph Reisinger, Kaushansky, Vadim +5
Computer Science · Mathematics · #35B44 #60H30 #65N20 #80A22 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2010.05281
openalex publication_date 2020/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.