Mathematical models and software based on them are developed to simulate the complex processes of structural-phase transformations in next-generation materials, such as phase change materials (PCMs), biomedical materials, materials for additive manufacturing, and materials for the space industry. A mathematical description is performed for the enthalpy–porosity model. To describe liquid motion in time and space, the hydrodynamical equations of a viscous liquid are used. The necessary constraints and assumptions, which are related to the consideration of laminar flows and the Newtonian liquid model, for the model are analyzed. A computational problem is formulated in terms of the finite volume method, the discretization of a computational domain, and the hydrodynamical equations. The OpenFOAM software, i.e., an open integrated platform for a numerical simulation of continuum mechanics problems, is used for calculations. It is used to build an OpenFOAM computational algorithm in order to analyze the physical state of the system with allowance for initial and boundary conditions in the cases of conductive and convective heat transfer. The melting of gallium is simulated and the model is verified for the conductive and convective cases. In the conductive case, the melting of the material is shown to occur uniformly along heat sources, whereas different convection flow velocities exert a significant effect on melting boundary formation. The developed mathematical models, the analytical dependences obtained using them, and the performed computer simulations can be used to describe the real experimental data on crystal growth in supersaturated solutions and supercooled melts.