Heat transfer with a moving boundary has wide industrial applications, such as casting of metals, manufacturing of ice, and energy storage with phase change. In all the applications, melting and solidification lead to movements of the interface which has to be predicted for evaluation of the heat transfer rate. The method, previously developed for one-dimensional melting and solidification, is tested for a two-dimensional geometry. It is based on the assumption that the phase change does not occur at a fixed temperature point, but within a range of temperatures, which is true for most phase change materials. The effect of the latent heat of melting is included by using an effective heat capacity within the melting range selected. It was possible to predict the experimentally measured interface motion in a two-dimensional geometry containing paraffin wax. The method was also used to simulate a case of successive solidification and melting, leading to the formation of two interfaces.