In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some interesting numerical simulations which demonstrate the equilibrium structures and the approach of the system to non-constant equilibrium patterns. The novel feature of these calculations is the linking of the small parameter in the system, , to the grid spacing, thereby producing solutions with approximate sharp interfaces. Similar ideas have been used by Caginalp and Sokolovsky [5]. A movie of these simulations may be found at http:www.math.cmu.edu/math/people/greenberg.html
Mots-clés : phase-field models, melting and solidification
@article{M2AN_2001__35_4_607_0,
author = {Greenberg, James M.},
title = {Estimates and computations for melting and solidification problems},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {607--630},
publisher = {EDP-Sciences},
volume = {35},
number = {4},
year = {2001},
mrnumber = {1862871},
zbl = {0987.35016},
language = {en},
url = {http://archive.numdam.org/item/M2AN_2001__35_4_607_0/}
}
TY - JOUR AU - Greenberg, James M. TI - Estimates and computations for melting and solidification problems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2001 SP - 607 EP - 630 VL - 35 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/item/M2AN_2001__35_4_607_0/ LA - en ID - M2AN_2001__35_4_607_0 ER -
%0 Journal Article %A Greenberg, James M. %T Estimates and computations for melting and solidification problems %J ESAIM: Modélisation mathématique et analyse numérique %D 2001 %P 607-630 %V 35 %N 4 %I EDP-Sciences %U http://archive.numdam.org/item/M2AN_2001__35_4_607_0/ %G en %F M2AN_2001__35_4_607_0
Greenberg, James M. Estimates and computations for melting and solidification problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 607-630. http://archive.numdam.org/item/M2AN_2001__35_4_607_0/
[1] , An analysis of a phase-field model of a free boundary. Arch. Rat. Mech. Anal. 92 (1986) 205-245. | Zbl
[2] , Stefan and Hele-Shaw type models as asymptotic limits of the phase field equation. Phys. Rev. A 39 (1989) 5887-5896. | Zbl
[3] , Phase field models and sharp interface limits: some differences in subtle situations. Rocky Mountain J. Math. 21 (1996) 603-616. | Zbl
[4] and, Phase field equations in the singular limit of sharp interface problems, in On the evolution of phase boundaries, IMA 43 (1990-1991) 1-28. | Zbl
[5] and, Phase field computations of single-needle crystals, crystal growth, and motion by mean curvature. SIAM J. Sci. Comput. 15 (1994) 106-126. | Zbl
[6] and, The phase-field method in the sharp-interface limit: A comparison between model potentials. J. Comp. Phys. 130 (1997) 256-265. | Zbl
[7] ,,, and, Phase-field models for anisotropic interfaces. Phys. Rev. E 48 (1993) 2016-2024.
[8] and, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43 (1990) 44-62. | Zbl
[9] and, On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Physica D 69 (1993) 107-113. | Zbl
[10] ,,,,, and, Thermodynamically-consistent phase-field models. Physica D 69 (1993) 189-200. | Zbl
[11] and, Algorithms for phase field computations of the dendritic operating state at large supercoolings. J. Comp. Phys. 127 (1996) 110-117. | Zbl
