Numerical algorithms based on finite difference methods for a phase field model are discussed. The model describes the solidification of a pure substance and consists of an energy balance equation coupled to an Allen-Cahn equation with an additional forcing term, the unknown fields being the temperature and an order parameter representing one of the phases. \par The algorithms to solve the parabolic equations are presented in detail and carefully investigated with respect to convergence and physical relevance. Up to an (in time) explicit procedure an implicit procedure is developed involving a linearization of nonlinear terms and a time-splitting method. Among other issues, numerical tests concern the conservation of the internal energy and the convergence of the unknown fields in the $L^2$ norm. \par Numerical simulations reveal that the critical radius of solid nuclei is correctly recovered by the model and that the interface thickness parameter, when chosen in the right range, does not significantly influence the propagation speed of the solid-liquid interface. Further simulations serve to relate the propagation speed to the Stefan number, which physical parameters as the surface energy and the undercooling enter, and to investigate the consequences of slightly changing the initial shape of the solid domain.