  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.
