THERMAL SCIENCE

International Scientific Journal

Svetislav Savović

,

James Caldwell

NUMERICAL SOLUTION OF STEFAN PROBLEM WITH TIME-DEPENDENT BOUNDARY CONDITIONS BY VARIABLE SPACE GRID METHOD

ABSTRACT
The variable space grid method based on finite differences is applied to the one-dimensional Stefan problem with time-dependent boundary conditions describing the solidification/melting process. The temperature distribution, the position of the moving boundary and its velocity are evaluated in terms of finite differences. It is found that the computational results obtained by the variable space grid method exhibit good agreement with the exact solution. Also the present results for temperature distribution are found to be more accurate compared to those obtained previously by the variable time step method.
KEYWORDS
Stefan problem, variable space grid method, finite differences
PAPER SUBMITTED: 2008-10-20
PAPER REVISED: 2008-10-25
PAPER ACCEPTED: 2008-10-28
DOI REFERENCE: https://doi.org/10.2298/TSCI0904165S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2009, VOLUME 13, ISSUE No. 4, PAGES [165 - 174]
REFERENCES
[1] Hill, J., One-Dimensional Stefan Problem: An Introduction, Longman Scientific and Technical, Harlow, Essex, UK, 1987
[2] Goodman, T. R., The Heat-Balance Integral Method and its Application to Problems Involving a Change of Phase, Trans. ASME, 80 (1958), 2, pp. 335-342
[3] Crank, J., Free and Moving Boundary Problems, Clarendon Press, Oxford, UK, 1984
[4] Asaithambi, N. S., On a Variable Time Step Method for the One-Dimensional Stefan Problem, Comput. Meth. Appl. Mech. Engg., 71 (1988), 1, pp. 1-13
[5] Asaithambi, N. S., A Variable Time-Step Galerkin Method for a One-Dimensional Stefan Problem, Appl. Math. Comput., 81 (1997), 2-3, pp. 189-200
[6] Kutluay, S., Bahadir, A. R. Özdes, A., The Numerical Solution of One-Phase Classical Stefan problem, J. Comput. Appl. Math., 81 (1997), 1, pp. 135-144
[7] Wood, A. S., A New Look at the Heat Balance Integral Method, Appl. Math. Modelling, 25 (2001), 10, pp. 815-824
[8] Esen, A., Kutluay, S., A Numerical Solution of the Stefan Problem with a Neumann-Type Boundary Condition by Enthalpy Method, Appl. Math. Comput., 148 (2004), 2, pp. 321-329
[9] Ang, W.T., A Numerical Method Based on Integro-Differential Formulation for Solving a One-Dimensional Stefan Problem, Numerical Methods for Partial Differential Equations, 24 (2008), 3, pp. 939-949
[10] Mennig, J., Özisik, M. N., Coupled Integral Equation Approach for Solving Melting or Solidification, Int. J. Mass Transfer, 28 (1985), 8, pp. 1481-1485
[11] Furzeland, R. M., A Comparative Study of Numerical Methods for Moving Boundary Problems, J. Inst. Maths. Appl., 59 (1980), 26, pp. 411-429
[12] Rizwann-Uddin, One-Dimensional Phase Change with Periodic Boundary Conditions, Num. Heat Transfer, Part A, 35 (1999), 4, pp. 361-372
[13] Savović, S., Caldwell, J., Finite Difference Solution of One-Dimensional Stefan Problem with Periodic Boundary Conditions, Int. J. Heat and Mass Transfer, 46 (2003), 15, pp. 2911-2916
[14] Caldwell, J., Kwan, Y. Y., Numerical Methods for One-Dimensional Stefan Problems, Commun. Numer. Meth. Engng., 20 (2004), 7, pp. 535-545
[15] Marshall, G., A Front Tracking Method for One-Dimensional Moving Boundary Problems, SIAM J. Sci. Stat. Comput., 7 (1986), 1, pp. 252-263
[16] Churchill, S. W., Gupta, J. P., Approximations for Conduction with Freezing or Melting, Int. J. Heat Mass Transfer, 20 (1977), 11, pp. 1251-1253
[17] Voller, V. R., Cross, M., Applications of Control Volume Enthalpy Methods in the Solution of Stefan Problems, in: Computational Techniques in Heat Transfer (Eds. R. W. Lewis, K. Morgan, J. A. Johnson, W. R. Smith), Pineridge Press Ltd., Mumbles, Swansea, UK, 1985
[18] Caldwell, J., Chan, C. C., Spherical Solidification by the Enthalpy Method and the Heat Balance Integral Method, Appl. Math. Model., 24 (2000), 1, pp. 45-53
[19] Caldwell, J., So, K. L., Numerical Solution of Stefan Problems Using the Variable Time Step Finite-Difference Method, J. Math. Sciences, 11 (2000), 2, pp. 127-138
[20] Caldwell, J., Savović, S., Numerical Solution of Stefan Problem by Variable Space Grid Method and Boundary Immobilisation Method, J. Math. Sciences, 13 (2002), 1, pp. 67-79
[21] Gupta, R. S., Kumar, D., A Modified Variable Time Step Method for the One-Dimensional Stefan Problem, Comput. Meth. Appl. Mech. Engng., 23 (1980), 1, pp. 101-108
[22] Murray, W. D., Landis, F., Numerical and Machine Solutions of Transient Heat Conduction Involving Melting or Freezing, J. Heat Transfer, 81 (1959), pp. 106-112
[23] Finn, W. D., Voroglu, E., Finite Element Solution of the Stefan Problem, in: The Mathematics of Finite Elements and Applications, MAFELAP 1978 (Ed. J. R. Whiteman), Academic Press, New York, USA, 1979
PDF VERSION [DOWNLOAD]
Numerical solution of Stefan problem with time-dependent boundary conditions by variable space grid method

© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence