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Numerical study of two ill-posed one-phase Stefan problems

Back, Julian M., McCue, Scott W., & Moroney, Timothy J. (2011) Numerical study of two ill-posed one-phase Stefan problems. The ANZIAM Journal, 52, C430-C446.

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Abstract

We treat two related moving boundary problems. The first is the ill-posed Stefan problem for melting a superheated solid in one Cartesian coordinate. Mathematically, this is the same problem as that for freezing a supercooled liquid, with applications to crystal growth. By applying a front-fixing technique with finite differences, we reproduce existing numerical results in the literature, concentrating on solutions that break down in finite time. This sort of finite-time blow-up is characterised by the speed of the moving boundary becoming unbounded in the blow-up limit. The second problem, which is an extension of the first, is proposed to simulate aspects of a particular two-phase Stefan problem with surface tension. We study this novel moving boundary problem numerically, and provide results that support the hypothesis that it exhibits a similar type of finite-time blow-up as the more complicated two-phase problem. The results are unusual in the sense that it appears the addition of surface tension transforms a well-posed problem into an ill-posed one.

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ID Code: 42507
Item Type: Journal Article
Keywords: ill-posed Stefan problem, superheated solid, finite-time blow-up, surface tension, method of lines
ISSN: 1446-8735 (online) 1446-1811 (print)
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Theoretical and Applied Mechanics (010207)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Past > Schools > Mathematical Sciences
Copyright Owner: Copyright 2011 Cambridge University Press
Deposited On: 25 Jul 2011 11:08
Last Modified: 01 May 2012 18:26

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