## THERMAL SCIENCE

International Scientific Journal

### TWO NUMERICAL METHODS FOR HEAT CONDUCTION PROBLEMS WITH PHASE CHANGE

**ABSTRACT**

A fixed-space-step method and a fixed-time-step method are presented, respectively, for solving the Stefan problems with time-dependent boundary conditions. The evolution of the moving interface and the temperature distribution in the phase change domain are simulated numerically by using two methods for melting in the half-plane and outward spherical solidification. Numerical experiment results show that the numerical results obtained from the two methods are in good agreement for the different test examples, and the two methods can be applied to solve Stefan problems in engineering practice.

**KEYWORDS**

PAPER SUBMITTED: 2017-04-27

PAPER REVISED: 2017-09-19

PAPER ACCEPTED: 2017-09-20

PUBLISHED ONLINE: 2018-09-10

**THERMAL SCIENCE** YEAR

**2018**, VOLUME

**22**, ISSUE

**Issue 4**, PAGES [1787 - 1794]

- Javierre, E., et al., A Comparison of Numerical Models for One-Dimensional Stefan Problems, Journal of Computational and Applied Mathematics, 192 (2006), 2, pp. 445-459
- Rajeev, Kushwaha, M. S., An Approximate Approach for a Stefan Problem with Periodic Boundary Condition, Journal of Engineering Computers & Applied Sciences, 1 (2012), 1, pp. 66-73
- Qu, L. H., et al., An Approximate Method for Solving Melting Problem with Periodic Boundary Condi-tions, Thermal Science, 18 (2014), 5, pp. 1679-1684
- Vrankar, L., et al., Moving-Boundary Problems Solved by Adaptive Radial Basis Functions, Computers & Fluids, 39 (2010), 9, pp. 1480-1490
- Tadi, M., A Four-Step Fixed-Grid Method for 1D Stefan Problems, Journal of Heat Transfer, 132 (2010), 11, pp. 114502.1-114502.4
- Mitchell, S. L., Myers, T. G., Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems, SIAM Review, 52 (2010), 1, pp. 57-86
- Caldwell, J., Kwan, Y. Y., Numerical Methods for One-Dimensional Stefan Problems, Communications in Numerical Methods in Engineering, 20 (2004), 7, pp. 535-545
- Qu, L. H., et al., Numerical Study of One-Dimensional Stefan Problem with Periodic Boundary Condi-tions, Thermal Science, 17 (2013), 5, pp. 1453-1458
- Mitchell, S. L., Vynnycky, M., Finite-Difference Methods with Increased Accuracy and Correct Initiali-zation for One-Dimensional Stefan Problems, Applied Mathematics and Computation, 215 (2009), 4, pp. 1609-1621
- Caldwell, J., Kwan, Y. Y., A Brief Review of Several Numerical Methods for One-Dimensional Stefan Problems, Thermal Science, 13 (2009), 2, pp. 61-72