论文标题
在存在爆破的情况下，超冷的Stefan问题的隐式和完全离散的近似
Implicit and fully discrete approximation of the supercooled Stefan problem in the presence of blow-ups
论文作者
论文摘要
我们考虑了一维超冷的Stefan问题的两个隐式近似方案，即使存在有限的时间爆炸，也证明了它们的收敛性。所有证据均基于文献中最近考虑的概率重新印象。第一个方案是在V. Kaushansky，C。Reisinger，M。Shkolnikov和Z. Q. Song，Arxiv：2010.05281，2020中研究的时间步长方案的版本，但是在这里，自由边界上的通量及其速度是隐含的。此外，我们将分析扩展到比布朗运动更多的一般驾驶过程。第二个方案是Donsker型近似，也可以解释为隐式有限差方案，在次要的技术条件下显示全局收敛。在没有爆炸的情况下适用的更强的假设，我们将收敛速率任意接近1/2。我们的数值结果表明，与显式方案相比，该速率也适用于较少的常规解决方案，并且可以在爆炸制度中对不连续的自由边界进行更清晰的分辨率。
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.
