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In the present paper a computational model for the macroscopic freezing mechanism under supercooled conditions relying on the physical and mathematical description of the two-phase Stefan problem is formulated. The relevant numerical algorithm based on the finite volume method is implemented into the open source software OpenFOAM©. For the numerical capturing of the moving interface between the supercooled and the solidified liquid an appropriate level set formulation is utilized. The heat transfer equations are solved in both the liquid phase and solid phase independently from each other. At the interface a Dirichlet boundary condition for the temperature field is imposed and a ghost-face method is applied to ensure accurate calculation of the normal derivative needed for the jump condition, i.e. for the interface-velocity in the normal direction. For the sake of updating the level set function a narrow-band around the interface is introduced. Within this band, whose width is temporally adjusted to the maximum curvature of the interface, the normal-to-interface velocity is appropriately expanded. The physical model and numerical algorithm are validated along with the analytical solution. Understanding instabilities is the first step in controlling them, so to quantify all sorts of instabilities at the solidification front the Mullins-Sekerka theory of morphological stability is investigated.