Numerical solutions of the moving boundary problems

This thesis consists of six chapters. In the first chapter, after giving finite difference and finite element methods with general lines, spline functions, B-spline bases functions and Von-Neumann stability analysis method are presented. In the second chapter, after introducing the Stefan problems generally, six problems which are considered with different initial and boundary conditions have been analyzed with their existing exact solutions. The third, fourth and fifth chapters of this thesis make up its original parts. In the third chapter, collocation finite element method based on variable space grid method is constructed using cubic B-spline bases functions. Numerical results for temperature distribution, the position and velocity of moving boundary are obtained by applying the method to six different problems. The obtained numerical results are compared with the exact solutions and the other numerical results existing in the literature and the error norms are given. Also stability analysis of generated finite element methods are investigated with Von-Neumann stability analysis method. In the fourth chapter, collocation finite element method based on boundary immobilisation method is constructed using cubic B-spline bases functions. Numerical results for temperature distribution, the position and velocity of moving boundary are obtained by applying the method to six different problems. The obtained numerical results are compared with the exact solutions and the other numerical results existing in the literature and the error norms are given. Also stability analysis of generated finite element methods are investigated with Von-Neumann stability analysis method. In the fifth chapter, collocation finite element method based on isotherm migration method is constructed using cubic B-splines. For the three problems which isotherm migration method can't be used directly, by using a suitable transformation, the method is applied to four different problems. Numerical solutions obtained for the locations of isotherms, the position and velocity of moving boundary are compared with the exact solutions and the other numerical results existing in the literature and the error norms are given. In the sixth chapter, the numerical results obtained in the third, fourth and fifth chapters are evaluated.