The crystallization mechanism of pure water in a supercooled state is not well understood so far. There are many open-ended questions about the basic physics of crystallization. A new computational model using an appropriate level set formulation for the numerical capturing of the interface between the supercooled and the solidified liquid is applied. Mathematically, the phenomenon of solidification is modeled by utilizing a moving boundary problem. Recent numerical results of dendritic growth (Criscione et al. 2012) exhibit excellent qualitative and quantitative agreement with the Marginal Stability Theory (Langer & Müller Krumbhaar 1978a, 1978b, 1978c) as well as with the available experiments (Furukawa & Shimada 1993, Ohsaka & Trinh 1998, Shibkov et al. 2001, 2003, 2005) in the heat-diffusion-dominated region. At higher supercoolings (in the so-called kinetics-limited region), an explicit deviation from experiments is observed. In the published literature the kinetic effects are indicated as a possible reason for this deviation, approximating the kinetic undercooling as a linear function of the interface velocity. Based on this assumption, a new approach for the calculation of the kinetic undercooling term is derived. This model results in an approximation for the kinetic coefficient which establishes a non-linear dependency between the kinetic undercooling and the velocity of the solid-liquid interface. Furthermore, investigations concerning the growth of needles in an array indicate that surrounding needle-like dendrites influence considerably the steady-state tip velocity of an isolated needle. This phenomenon depends directly on the spacing between the needles. In the present work an attempt is undertaken to explain a new approach for the physical description of the crystallization mechanism at higher supercooling.