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

## DOI:

https://doi.org/10.21914/anziamj.v52i0.3937## Keywords:

Stefan problem, size-dependent melting, surface tension, finite-time blow-up, superheating## 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, 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.**References**

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## Published

2011-07-23

## Issue

## Section

Proceedings Computational Techniques and Applications Conference