Numerical solution of Stefan problem with variable space grid method based on mixed finite element/finite difference approach

Abstract

The purpose of this paper is to improve the accuracy and stability of the existing solutions to 1D Stefan problem with time-dependent Dirichlet boundary conditions. The accuracy improvement should come with respect to both temperature distribution and moving boundary location. The variable space grid method based on mixed finite element/finite difference approach is applied on 1D Stefan problem with time-dependent Dirichlet boundary conditions describing melting process. The authors obtain the position of the moving boundary between two phases using finite differences, whereas finite element method is used to determine temperature distribution. In each time step, the positions of finite element nodes are updated according to the moving boundary, whereas the authors map the nodal temperatures with respect to the new mesh using interpolation techniques. The authors found that computational results obtained by proposed approach exhibit good agreement with the exact solution. Moreover, the results for temperature distribution, moving boundary location and moving boundary speed are more accurate th

an those obtained by variable space grid method based on pure finite differences. The authors’ approach clearly differs from the previous solutions in terms of methodology. While pure finite difference variable space grid method produces stable solution, the mixed finite element/finite difference variable space grid scheme is significantly more accurate, especially in case of high alpha. Slightly modified scheme has a potential to be applied to 2D and 3D Stefan problems.