Estimates and computations for melting and solidification problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, p. 607-630

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, δ, 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

Classification:  35B25,  35B40,  35B45
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {4},
     year = {2001},
     pages = {607-630},
     zbl = {0987.35016},
     mrnumber = {1862871},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_4_607_0}
}
Greenberg, James M. Estimates and computations for melting and solidification problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, pp. 607-630. http://www.numdam.org/item/M2AN_2001__35_4_607_0/

[1] G. Caginalp, An analysis of a phase-field model of a free boundary. Arch. Rat. Mech. Anal. 92 (1986) 205-245. | Zbl 0608.35080

[2] G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equation. Phys. Rev. A 39 (1989) 5887-5896. | Zbl 1027.80505

[3] G. Caginalp, Phase field models and sharp interface limits: some differences in subtle situations. Rocky Mountain J. Math. 21 (1996) 603-616. | Zbl 0753.35125

[4] G. Caginalp and X. Chen, 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 0760.76094

[5] G. Caginalp and E. Sokolovsky, Phase field computations of single-needle crystals, crystal growth, and motion by mean curvature. SIAM J. Sci. Comput. 15 (1994) 106-126. | Zbl 0793.65099

[6] M. Fabbri and V.R. Vollmer, The phase-field method in the sharp-interface limit: A comparison between model potentials. J. Comp. Phys. 130 (1997) 256-265. | Zbl 0868.65094

[7] G.B. Mcfadden, A.A. Wheeler, R.J. Brown, S.R. Coriell and R.F. Sekerka, Phase-field models for anisotropic interfaces. Phys. Rev. E 48 (1993) 2016-2024.

[8] O. Penrose and P. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43 (1990) 44-62. | Zbl 0709.76001

[9] O. Penrose and P. Fife, On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Physica D 69 (1993) 107-113. | Zbl 0799.76084

[10] S.L. Wang, R.F. Sekerka, A.A. Wheeler, B.T. Murray, S.R. Coriell, R.J. Braun and G.B. Mcfadden, Thermodynamically-consistent phase-field models. Physica D 69 (1993) 189-200. | Zbl 0791.35159

[11] S.L. Wang and R.F. Sekerka, Algorithms for phase field computations of the dendritic operating state at large supercoolings. J. Comp. Phys. 127 (1996) 110-117. | Zbl 0859.65131