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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
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 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

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