A semi-analytic numerical scheme has been developed to solve the one-dimensional, moving boundary phase change problem with time-dependent boundary conditions. Locally analytic, approximate solutions are developed for the position of the moving boundary, and for temperature distribution. Set of discrete equations are obtained by applying these solutions over space-time nodes, and by imposing continuity of temperature and heat flux. Application of this so-called nodal integral approach to the nonlinear Stefan problem shows that the scheme isO(Δx2), and that it predicts the position of the moving boundary and the temperature distribution within the domain very accurately. For example, with as little as two nodes in the spatial domain, the location of the moving boundary for the case of an exponentially increasing surface temperature on the boundary, after one dimensionless time unit, is found with an error of less than 1%. In addition to large size nodes in space, this scheme also allows the use of very large size time steps. Comparison of numerical results with reference solutions is presented.