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Godunov Method for Stefan Problems with Enthalpy Formulations

Published online by Cambridge University Press:  28 May 2015

D. Tarwidi  and
S.R. Pudjaprasetya
D. Tarwidi*
Affiliation:
Department of Computational Science, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia
S.R. Pudjaprasetya*
Affiliation:
Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia
*
Corresponding author. Email: dede.tarwidi@yahoo.com
Corresponding author. Email: sr_pudjap@math.itb.ac.id
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Abstract

A Stefan problem is a free boundary problem where a phase boundary moves as a function of time. In this article, we consider one-dimensional and two-dimensional enthalpy-formulated Stefan problems. The enthalpy formulation has the advantage that the governing equations stay the same, regardless of the material state (liquid or solid). Numerical solutions are obtained by implementing the Godunov method. Our simulation of the temperature distribution and interface position for the one-dimensional Stefan problem is validated against the exact solution, and the method is then applied to the two-dimensional Stefan problem with reference to cryosurgery, where extremely cold temperatures are applied to destroy cancer cells. The temperature distribution and interface position obtained provide important information to control the cryosurgery procedure.

Keywords

65M1078A48Stefan problemsGodunov methodsolidificationenthalpycryosurgery
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 2 , May 2013 , pp. 107 - 119
Copyright
Copyright © Global-Science Press 2013

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References

[1]Alexiades, V. and Solomon, A.D., Mathematical Modelling of Melting and Freezing Processes, Hemisphere Publishing Corporation, Washington DC, 1981.Google Scholar
[2]Caldwell, J. and Kwan, Y.Y., Numerical methods for one-dimensional Stefan problems, Communications in Numerical Methods in Engineering, 20 (2004), pp. 535–545.Google Scholar
[3]Voller, V.R. and Shadabi, L., Enthalpy methods for tracking a phase change boundary in two dimensions, International Communications in Heat and Mass Transfer, 11 (1984), pp. 239–249.Google Scholar
[4]Voller, V. and Cross, M., Accurate solutions of moving boundary problems using the enthalpy method, International Journal of Heat and Mass Transfer, 24 (1981), pp. 545–556.CrossRefGoogle Scholar
[5]Esen, A. and Kutluay, S., A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method, Applied Mathematics and Computation, 148 (2004), pp. 321–329.Google Scholar
[6]Leveque, R.J., Finite-Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
[7]Toro, E.F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer-Verlag, Berlin, Heidelberg, 2009.Google Scholar
[8]Wendroff, B., Approximate Riemann solvers, Godunov schemes and contact discontinuities, in Godunov Methods: Theory and Applications, Toro, E.F. (ed.), Kluwer Academic/Plenum Publishers, New York, 2001.Google Scholar
[9]Colella, P., Volume-of-fluid methods for partial differential equations, in Godunov Methods: Theory and Applications, Toro, E.F. (ed.), Kluwer Academic/Plenum Publishers, New York, 2001.Google Scholar
[10]Leer, B. Van, Upwind and high-resolution methods for compressible flow: From donor cell to residual-distribution schemes, Communication in Computational Physics, 1 (2006), pp. 192–206.Google Scholar
[11]Zhao, G., et. al, Comparative study of the cryosurgical processes with two different cryosurgical systems: the endocare cryoprobe system versus the novel combined cryosurgeryand hyperthermia system, Latin American Applied Research, 37 (2007), pp. 215–222.Google Scholar
[12]Kumar, S. and Katiyar, V.K., Numerical study on phase change heat transfer during combined hyperthermia and cryosurgical treatment of lung cancer, International Journal of Applied Mathematics and Mechanics, 3 (2007), pp. 1–17.Google Scholar