A modified finite difference scheme for the Stefan problem
Author:
R. E. White
Journal:
Math. Comp. 41 (1983), 337347
MSC:
Primary 65M10; Secondary 80A20
MathSciNet review:
717690
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Abstract 
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Abstract: In this paper we describe an algorithm which can be used to approximate the solution to the enthalpy formulation of the Stefan problem. We allow the thermal properties to have a space and temperature dependence. The algorithm is not explicit in the time variable and, hence, the stability condition on is not too severe. A proof of convergence is given and two numerical examples are presented.
 [1]
D.
R. Atthey, A finite difference scheme for melting problems, J.
Inst. Math. Appl. 13 (1974), 353–366. MR 0351295
(50 #3784)
 [2]
Gunter
H. Meyer, Multidimensional Stefan problems, SIAM J. Numer.
Anal. 10 (1973), 522–538. MR 0331807
(48 #10139)
 [3]
J. Ockendon & W. Hodgkins (Editors), Moving Boundary Value Problems in Heat Flow and Diffusion, Clarendon Press, Oxford, 1975.
 [4]
J.
M. Ortega and W.
C. Rheinboldt, Iterative solution of nonlinear equations in several
variables, Academic Press, New YorkLondon, 1970. MR 0273810
(42 #8686)
 [5]
Milton
E. Rose, A method for calculating solutions of
parabolic equations with a free boundary, Math.
Comput. 14 (1960),
249–256. MR 0115283
(22 #6085), http://dx.doi.org/10.1090/S00255718196001152838
 [6]
L.
I. Rubenstein, The Stefan problem, American Mathematical
Society, Providence, R.I., 1971. Translated from the Russian by A. D.
Solomon; Translations of Mathematical Monographs, Vol. 27. MR 0351348
(50 #3837)
 [7]
N. Shamsundar & E. M. Sparrow, "Analysis of multidimensional conduction phase change via the enthalpy model," J. Heat Transfer, v. 97, 1975, pp. 333340.
 [8]
Alan
Solomon, Some remarks on the Stefan
problem, Math. Comp. 20 (1966), 347–360. MR 0202391
(34 #2262), http://dx.doi.org/10.1090/S00255718196602023911
 [9]
A. Solomon, "Melt time and heat flux for a simple PCM body," Solar Energy, v. 22, 1979, pp. 251257.
 [10]
A. Solomon, On the Use of Mathematical Methods in Cryosurgery, Union Carbide Corp., Nuclear Div., Computer Sci. Div., Oak Ridge National Laboratory, Oak Ridge, Tenn., May 1980.
 [11]
J. Stefan, "Über einige Probleme der Theorie de Warmeleitung," S. Ber. Wieh. Akad. Mat. Natur., v. 98, 1889, pp. 473484.
 [12]
J. A. Wheeler, "Permafrost thermal design for transAlaskan pipeline," in Moving Boundary Problems, Academic Press, New York, 1978.
 [13]
R.
E. White, An enthalpy formulation of the Stefan problem, SIAM
J. Numer. Anal. 19 (1982), no. 6, 1129–1157. MR 679656
(84h:65092a), http://dx.doi.org/10.1137/0719082
 [14]
R.
E. White, A numerical solution of the enthalpy formulation of the
Stefan problem, SIAM J. Numer. Anal. 19 (1982),
no. 6, 1158–1172. MR 679657
(84h:65092b), http://dx.doi.org/10.1137/0719083
 [15]
David
George Wilson, Alan
D. Solomon, and Paul
T. Boggs (eds.), Moving boundary problems, Academic Press
[Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1978. MR 0466887
(57 #6761)
 [1]
 D. R. Atthey, "A finite difference scheme for melting problems," J. Inst. Math. Appl., v. 13, 1974, pp. 353366. MR 0351295 (50:3784)
 [2]
 G. H. Meyer, "Multidimensional Stefan problems," SIAM J. Numer. Anal., v. 10, 1973, pp. 522538. MR 0331807 (48:10139)
 [3]
 J. Ockendon & W. Hodgkins (Editors), Moving Boundary Value Problems in Heat Flow and Diffusion, Clarendon Press, Oxford, 1975.
 [4]
 J. M. Ortega & W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
 [5]
 M. Rose, "A method for calculating solutions of parabolic problems with a free boundary." Math. Comp., v. 14, 1960, pp., 249256. MR 0115283 (22:6085)
 [6]
 L. I. Rubinstein, The Stefan Problem, Transl. Math. Mono., vol. 27, Amer. Math. Soc., Providence, R. I., 1971. MR 0351348 (50:3837)
 [7]
 N. Shamsundar & E. M. Sparrow, "Analysis of multidimensional conduction phase change via the enthalpy model," J. Heat Transfer, v. 97, 1975, pp. 333340.
 [8]
 A. Solomon, "Some remarks on the Stefan problem," Math. Comp., v. 20, 1966, pp. 347360. MR 0202391 (34:2262)
 [9]
 A. Solomon, "Melt time and heat flux for a simple PCM body," Solar Energy, v. 22, 1979, pp. 251257.
 [10]
 A. Solomon, On the Use of Mathematical Methods in Cryosurgery, Union Carbide Corp., Nuclear Div., Computer Sci. Div., Oak Ridge National Laboratory, Oak Ridge, Tenn., May 1980.
 [11]
 J. Stefan, "Über einige Probleme der Theorie de Warmeleitung," S. Ber. Wieh. Akad. Mat. Natur., v. 98, 1889, pp. 473484.
 [12]
 J. A. Wheeler, "Permafrost thermal design for transAlaskan pipeline," in Moving Boundary Problems, Academic Press, New York, 1978.
 [13]
 R. E. White, "An enthalpy formulation of the Stefan problem," SIAM J. Numer. Anal., v. 19, 1982, pp. 11291157. MR 679656 (84h:65092a)
 [14]
 R. E. White, "A numerical solution of the enthalpy formulation of the Stefan problem," SIAM J. Numer. Anal., v. 19, 1982, pp. 11581173. MR 679657 (84h:65092b)
 [15]
 D. G. Wilson, A Solomon & P. T. Boggs (Editors), Moving Boundary Problems, Academic Press, New York, 1978. MR 0466887 (57:6761)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307176904
PII:
S 00255718(1983)07176904
Article copyright:
© Copyright 1983
American Mathematical Society
