SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1145-1170, June 2024.
Abstract.We consider two 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 by Kaushansky et al. [Ann. Appl. Probab., 33 (2023), pp. 274–298], 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.

中文翻译：

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存在爆炸时过冷 Stefan 问题的隐式和完全离散逼近

SIAM 数值分析杂志，第 62 卷，第 3 期，第 1145-1170 页，2024 年 6 月。
摘要。我们考虑一维过冷 Stefan 问题的两种近似方案，并证明它们的收敛性，即使存在有限时间爆炸。所有证明均基于文献中最近考虑的概率重新表述。第一个方案是 Kaushansky 等人研究的时间步进方案的一个版本。 [安。应用。 Probab., 33 (2023), pp. 274–298]，但这里自由边界上的通量与其速度是隐式耦合的。此外，我们将分析扩展到比布朗运动更一般的驱动过程。第二种方案是Donsker型近似，也可以解释为隐式有限差分方案，在较小的技术条件下显示出全局收敛性。有了更强的假设（适用于没有爆炸的情况），我们还获得了任意接近 1/2 的收敛率。我们的数值结果表明，与显式方案相比，该速率也适用于不太规则的解决方案，并且允许更清晰地解析爆炸状态中的不连续自由边界。