Implicit and Fully Discrete Approximation of the Supercooled Stefan Problem in the Presence of Blow-Ups

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-05-09 DOI:10.1137/22m1509722

Christa Cuchiero, Christoph Reisinger, Stefan Rigger

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引用次数: 0

Abstract

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.

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存在炸裂的过冷斯特凡问题的隐含和完全离散近似法

SIAM 数值分析期刊》第 62 卷第 3 期第 1145-1170 页，2024 年 6 月。摘要.我们考虑了一维过冷斯特凡问题的两种近似方案，并证明了它们的收敛性，即使在存在有限时间炸裂的情况下也是如此。所有证明都基于最近文献中考虑的概率重述。第一个方案是 Kaushansky 等人研究的时间步进方案的一个版本[Ann. Appl. Probab.此外，我们还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，在一些次要的技术条件下，可以显示全局收敛性。通过更强的假设（适用于没有炸毁的情况），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，该收敛率也适用于不太规则的解，并能更清晰地解决炸毁机制中的不连续自由边界问题。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.