## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### A parameters selection criterion of the numerical realisation of the continuous method for the Stefan problems

**ABSTRACT**

This paper suggests a selection criterion of the continuous method version for a numerical solution of the Stefan problem which would allow to calculate the phase transition boundary position with a required accuracy for a long period of time and would enable generalisation to multidimensional problems. Despite a large number of works deal with the solution to the generalised Stefan problem by the continuous method, the choice of the smoothing interval value for numerical feasibility is not fully clear. A comparison of the calculation accuracy of the phase transition boundary position using different versions of the continuous method was carried out on an example of the well known one-dimensional plane two-phase Stefan problem which possesses an analytical solution. The dependence of the total error of the numerical calculation of the phase transition boundary position on the value of the smearing interval is determined from the comparison of numerical and analytical solutions. An analysis of the reason for increase of this error with time at any choice of a constant smoothing interval is given. A version of the continuous method with a variable interval of the delta function smoothing, in which the proposed criterion is carried out, is discussed. The position of the phase transition boundary calculated proposed version matches the analytical solution with a required accuracy over a long period of time.

**KEYWORDS**

PAPER SUBMITTED: 2019-01-21

PAPER REVISED: 2019-07-08

PAPER ACCEPTED: 2019-07-11

PUBLISHED ONLINE: 2019-08-10

- Douglas, J., Gallie, T. M., On the numerical integration of a parabolic differential equation subject to a moving boundary condition, Duke Math. J. 22 (1955), pp. 557-70.
- Vasil'ev, F. P., On finite difference methods for the solution of Stefan's single- phase problem, USSR Comput. Math. Math. Phys. 3 (1963), 5 pp. 1175-1191.
- Crank J., Free and moving boundary problems. Oxford University Press, New York, (1984).
- Samarskii, A. A., Moiseyenko, B.D., An economic continuous calculation scheme for the Stefan multi-dimensional problems, USSR Comput. Math. Math. Phys. 5 (1965), 5, pp. 43-58.
- Budak, B. M., Sobol'eva, E.N., Uspenskii, A.B., A difference method with coefficient smoothing for the solution of Stefan problems, USSR Comput. Math. Math. Phys. 5 (1965), 5, pp. 59-76.
- Kamenomostskaja, S. L., On Stefan's problem, Mat. Sb. 53 (1965), 95, pp. 485-514.
- Oleinik, O. A., On a method of solution of the general Stefan problem, Dokl. AN SSSR, 135 (1960), 5, pp. 1054-1057.
- Friedman, A., The Stefan problem in several space variables, Trans. Am. Math. Soc. 133, (1968), pp. 51- 87.
- Eyres, N.R., Hartree, D.R., Ingham, J. et al, The calculation of variable heat flow in solids, Philosophical Transactions of the Royal Society. Series A., 240 (1946), pp. 1-57.
- Caldwell, J., Chan, C., Numerical solution of the Stefan problem by the enthalpy method and the heat balance integral method, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 33 (1998), pp. 99-117.
- Azaiez, M., Jelassi, F., Brahim, M. et al, Two phases of Stefan problem with smoothed enthalpy, Commun. Math. Sci., 14 (2016), 6, pp. 1625-1641.
- Feulvarch, E., Bergheau, J.M., Leblond, J.B., An implicit finite element algorithm for the simulation of diffusion with phase changes in solids, Int. J. Num. Methods Eng. 78 (2009), 12, pp. 1492-1512.
- Voller, V.R., An implicit enthalpy solution for phase change problems: with application to a binary alloy solidification, Appl. Math. Modelling 11 (1987), pp. 110-116.
- Krivovichev, G.V., A computational approach to the modeling of the glaciation of sea offshore gas pipeline, International Journal of Heat and Mass Transfer, 115 (2017), pp. 1132-1148.
- Mikova, V.V., Kurbatova, G.I., Ermolaeva, N.N., Analytical and numerical solutions to Stefan problem in model of the glaciation dynamics of the multilayer cylinder in sea water. Journal of Physics: Conf. Series, 2017. (929) 012103.
- Tichonov, A.N., Samarskii, A.A., Equations of mathematical physics, Pergamon Press Ltd., USA, 1963.
- Budak, B.M., Vasilev, F.P., Uspenskii, A.B., Difference methods of the solution some boundary Stefan problems, VC MGU, Chislennie metodi v gasovoj dinamiki, (1965) pp. 139 - 183.
- Shamsundar, N., Sparrow, E. M., Analysis of Multidimensional Conduction Phase Change Via the Enthalpy Model. J. Heat Transfer, vol. 97 (1975), 3, pp. 333 - 340.
- Vasilev, V.I., Maksimov, A.M., Petov, E.E.et al The heat and mass transfer in the freezing ground. Nauka Physmatlit Publ., Moscow, 1996.