Numerical study of two ill-posed one phase Stefan problems

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 S. D. Howison, J. R. Ockendon, and A. A. Lacey. Singularity development in moving-boundary problems. Q. J. Mech. Appl. Math., 38(3):343--360, 1985. doi:10.1093/qjmam/38.3.343 A. A. Lacey and J. R. Ockendon. Ill-posed free boundary problems. Control and Cybernetics, 14:275--296, 1985. A. Fasano and M. Primicerio. General free-boundary problems for the heat equation I. J. Math. Anal. Appl, 57:694--723, 1977. A. Fasano, M. Primicerio, and A. A. Lacey. New results on some classical parabolic free-boundary problems. Quart. Appl. Math., 38:439--460, 1981. M. A. Herrero and J. J. L. Velazquez. Singularity formation in the one-dimensional supercooled Stefan problem. Euro. J. Appl. Math., 7:119--150, 1996. doi:10.1137/04060528X B. Sherman. A general one-phase Stefan problem. Quart. Appl. Math., 28:377--382, 1970. J. R. King and J. D. Evans. Regularization by kinetic undercooling of blow-up in the ill-posed Stefan problem. SIAM J. Appl. Math., 65(5):1677--1707, 2005. doi:10.1137/04060528X J. Crank. Free and moving boundary problems. Clarendon Press, Oxford, 1984. T. C. Illingworth and I. O. Golosnoy. Numerical solutions of diffusion-controlled moving boundary problems which conserve solute. J. Comp. Phys., 209:207--225, 2005. doi:10.1016/j.jcp.2005.02.031 F. Liu and D. L. S. McElwain. A computationally efficient solution technique for moving-boundary problems in finite media. IMA J. Appl. Math, 59:71--84, 1997. doi:10.1093/imamat/59.1.71 S. W. McCue, B. Wu, and J. M. Hill. Classical two-phase Stefan problem for spheres. Proc. R. Soc. A, 464:2055--2076, 2008. doi:10.1098/rspa.2007.0315 S. W. McCue, B. Wu, and J. M. Hill. Micro/nanoparticle melting with spherical symmetry and surface tension. IMA J. Appl. Math., 74:439--457, 2009. doi:10.1093/imamat/hxn038 B. Wu, S. W. McCue, P. Tillman, and J. M. Hill. Single phase limit for melting nanoparticles. Appl. Math. Model., 33:2349--2367, 2009. doi:10.1016/j.apm.2008.07.009 H. G. Landau. Heat conduction in a melting solid. Quart. J. Appl. Math., 8:81--94, 1950. L. F. Shampine and M. W. Reichelt. The MATLAB ODE Suite. SIAM Journal on Scientific Computing, 18:1--22, 1997. S. C. Gupta. The classical Stefan problem: Basic concepts, modelling and analysis. Elsevier, Amsterdam, 2003.