Numerical solutions for a Stefan problem

Öz

The initial version of a Stefan problem is the melting of a semi-infinite sheet of ice. This problem is described by a parabolic partial differential equation along with two boundary conditions on the moving boundary which are used to determine the boundary itself and complete the solution of the differential equation. In this paper firstly, we use variable space grid method, boundary immobilisation method and isotherm migration method to get rid of the trouble of the Stefan problem. Then, collocation finite element method based on cubic B-spline bases functions is applied to model problem. The numerical schemes of finite element methods provide a good numerical approximation for the model problem. The numerical results show that the present results are in good agreement with the exact ones.

Anahtar Kelimeler

• Stefan problems,variable space grid method,boundary immobilisation method,isotherm migration method

Kaynakça

1. J. Stefan, Uber die theorie der eisbildung inbesondee uber die eisbindung im polarmeere, Ann. Phys. U. Chem. 42 (1891) 269-286.
2. J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984.
3. W.D. Murray, and F. Landis, Numerical and Machine Solutions of Transient Heat Conduction Involving Melting or Freezing, J. Heat Transfer 81 106-112 (1959).
4. S. Kutluay, A.R. Bahadir, A. Ozdes, The numerical solution of one-phase classical Stefan problem, J. Comp. Appl. Math. 81 (1997) 35-44.
5. N. S. Asaithambi, A Variable Time-Step Galerkin Method for the One- Dimensional Stefan Problem, Applied Mathematics and Computation 81 (1997) 189-200.
6. T.R.Goodman, The Heat-Balance Integral and its Application to Problems Involving a Change of Phase, Trans. ASME 80 (1959) 335-342.
7. H.G. Landau, Heat conduction in a melting solid, Quart. J. Appl. Math. 8 (1950) 81-94.
8. F.L. Chernousko,, Solution of non-linear Problems in Medium with Changes, Int. Chem. Engng. 10 (1970) 42-48.

9. R.C. Dix and J. Cizek, The isotherm migration method for transient heat conduction analysis, in: U. Grigull and E. Hahne, Eds., Heat Transfer 1, Proc. 4th Znternat. Heat Transfer Conf, Paris (Elsevier, Amsterdam, 1970) Cul.1.
10. A. Esen and S.Kutluay, An isotherm migration formulation for one-phase Stefan problem with a time dependent Neumann condition, Applied Mathematics and Computation 150 (2004) 59-67.
11. A. Esen, S. Kutluay, A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method, Applied Mathematics and Computation 148 (2004) 321-329.
12. J. Caldwell, S. Savovic, Y.Y. Kwan, Nodal Integral and Finite Difference Solution of One-Dimensional Stefan Problem, J. Heat Mass Trans. 125 (2003) 523-527.
13. A.H.A. Ali, G.A. Gardner, L.R.T. Gardner, A collocation solution for Burgers equation using cubic B-spline finite elements, Comput. Meth. Appl. Mech. Eng. 100 (1992) 325-337.
14. R.M. Furzeland, A comparative study of numerical methods for moving boundary problems, J. Inst. Maths. Appl. 26 (1980) 411-429.