Implicit and fully discrete approximation of the supercooled Stefan problem in the presence of blow-ups

TL;DR

This paper develops and proves convergence of two implicit numerical schemes for the one-dimensional supercooled Stefan problem, effectively handling finite time blow-ups and extending previous probabilistic reformulations.

Contribution

It introduces two novel implicit approximation schemes for the supercooled Stefan problem, including a flux-coupled time-stepping scheme and a Donsker-type scheme, with proven convergence even during blow-ups.

Findings

Convergence of the schemes is proven even with finite time blow-ups.

The Donsker-type scheme achieves near 1/2 convergence rate under certain conditions.

Numerical results indicate the schemes resolve discontinuities better than explicit methods.

Abstract

We consider two implicit approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme studied in V. Kaushansky, C. Reisinger, M. Shkolnikov, and Z. Q. Song, arXiv:2010.05281, 2020, but here the flux over the free boundary and its velocity are coupled implicitly. Moreover, we extend the analysis to more general driving processes than Brownian motion. The second scheme is a Donsker-type approximation, also interpretable as an implicit finite difference scheme, for which global convergence is shown under minor technical conditions. With stronger assumptions, which apply in cases without blow-ups, we obtain additionally a convergence rate arbitrarily close to 1/2. Our numerical results suggest that this rate also holds for less regular solutions, in contrast to explicit schemes, and allow a sharper resolution of the discontinuous free boundary in the blow-up regime.