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

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