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

Vadim Kaushansky, Christoph Reisinger, Mykhaylo Shkolnikov, Zhuo Qun Song

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)274-298
Number of pages25
JournalAnnals of Applied Probability
Volume33
Issue number1
DOIs
StatePublished - Feb 2023

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Blow-ups
  • free boundary
  • global Skorokhod M1 convergence
  • local convergence rates
  • particle approximation
  • physical solutions
  • probabilistic solutions
  • supercooled Stefan problem
  • time-stepping scheme

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