# Estimates and computations for melting and solidification problems

- Volume: 35, Issue: 4, page 607-630
- ISSN: 0764-583X

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topGreenberg, James M.. "Estimates and computations for melting and solidification problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.4 (2001): 607-630. <http://eudml.org/doc/194065>.

@article{Greenberg2001,

abstract = {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, $\delta $, 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},

author = {Greenberg, James M.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {phase-field models; melting and solidification},

language = {eng},

number = {4},

pages = {607-630},

publisher = {EDP-Sciences},

title = {Estimates and computations for melting and solidification problems},

url = {http://eudml.org/doc/194065},

volume = {35},

year = {2001},

}

TY - JOUR

AU - Greenberg, James M.

TI - Estimates and computations for melting and solidification problems

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2001

PB - EDP-Sciences

VL - 35

IS - 4

SP - 607

EP - 630

AB - 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, $\delta $, 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

LA - eng

KW - phase-field models; melting and solidification

UR - http://eudml.org/doc/194065

ER -

## References

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- [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. Zbl0709.76001
- [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. Zbl0799.76084
- [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. Zbl0791.35159
- [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. Zbl0859.65131

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