Variable space grid and boundary immobilisation schemes based on the explicit finite difference method are applied to the one-phase Stefan-like problems with a forcing term in order to evaluate the temperature distribution and the interface movement (location and speed). The numerical results obtained by the two schemes have been compared with the exact values and are found to be in good agreement with each other. It is shown that the numerical solutions exhibit the expected convergence to the exact one as the mesh size is reduced. A von-Neumann stability analysis of each scheme is also investigated. (c) 2004 Elsevier Inc. All rights reserved.