  We investigate the one-dimensional growth of a solid into a liquid bath,
starting from a small crystal, using the Guyer-Krumhansl and Maxwell-Cattaneo
models of heat conduction. By breaking the solidification process into the
relevant time regimes we are able to reduce the problem to a system of two
coupled ordinary differential equations describing the evolution of the
solid-liquid interface and the heat flux. The reduced formulation is in good
agreement with numerical simulations. In the case of silicon, differences
between classical and non-classical solidification kinetics are relatively
small, but larger deviations can be observed in the evolution in time of the
heat flux through the growing solid. From this study we conclude that the heat
flux provides more information about the presence of non-classical modes of
heat transport during phase-change processes.
