CONVERGENCE OF A TIME-STEPPING SCHEME TO THE FREE BOUNDARY IN THE SUPERCOOLED STEFAN PROBLEM

• 成果类型: Article
• 署名作者: Kaushansky, Vadim; Reisinger, Christoph; Shkolnikov, Mykhaylo; Song, Zhuo Qun
• 署名单位: University of California System; University of California Los Angeles; University of Oxford; Princeton University; Princeton University; Princeton University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1815
• 发表日期: 2023
• 页码: 274-298
• 关键词: kinetic condition; singular interaction; systems; MODEL

摘要

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.