Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem

TL;DR

This paper proves that an Euler time-stepping scheme converges to the free boundary in the supercooled Stefan problem, providing explicit convergence rates and supporting results with numerical tests.

Contribution

It establishes the global convergence of a natural numerical scheme to the free boundary in the supercooled Stefan problem, including explicit local convergence bounds.

Findings

Proves convergence of Euler scheme to the free boundary in the supercooled Stefan problem.

Provides explicit bounds on the local convergence rate.

Numerical tests confirm theoretical convergence behavior.

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.