Conduction and advection heat transfer in the solid and liquid metal, respectively, the free surface how of the liquid melt and its expulsion, the tracking of the solid-liquid and liquid-vapor interfaces with different thermo-physical properties in the two phases and the evolution of latent heat of fusion over a temperature range are mathematically modeled for the two-dimensional axisymmetric case in the transient development of a laser drilled hole where the impressed pressure and temperature on the melt surface is provided by a one-dimensional gas dynamics model. Significant improvement made to our earlier melting and solidification submodel is discussed that comprises a temperature transforming model on a fixed grid system. The resulting advection-diffusion equation’s compatibility with the present fluid flow simulation model is described. The mathematical formulation of the submodels and the numerical methodology is presented.