February 2023 Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem
Vadim Kaushansky, Christoph Reisinger, Mykhaylo Shkolnikov, Zhuo Qun Song
Author Affiliations +
Ann. Appl. Probab. 33(1): 274-298 (February 2023). DOI: 10.1214/22-AAP1815


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.

Funding Statement

M. Shkolnikov has been partially supported by the NSF Grants DMS-1811723, DMS-2108680 and a Princeton SEAS innovation research grant.


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Vadim Kaushansky. Christoph Reisinger. Mykhaylo Shkolnikov. Zhuo Qun Song. "Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem." Ann. Appl. Probab. 33 (1) 274 - 298, February 2023. https://doi.org/10.1214/22-AAP1815


Received: 1 October 2020; Revised: 1 September 2021; Published: February 2023
First available in Project Euclid: 21 February 2023

MathSciNet: MR4551550
zbMATH: 1515.80006
Digital Object Identifier: 10.1214/22-AAP1815

Primary: 35B44 , 65N20 , 80A22
Secondary: 60H30

Keywords: blow-ups , free boundary , global Skorokhod M1 convergence , local convergence rates , particle approximation , physical solutions , probabilistic solutions , supercooled Stefan problem , time-stepping scheme

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 1 • February 2023
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