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In this work, we study the liquid-solid interface dynamics for large time intervals on a 1-D sample, with homogeneous Neumann boundary conditions. In this kind of boundary value problem, we are able to make new predictions about the interface position by using conservation of energy. These predictions are confirmed through the heat balance integral method of Goodman and a generalized non-classical finite difference scheme. Since Neumann boundary conditions imply that the specimen is thermally isolated, through well stablished thermodynamics, we show that the interface behavior is not parabolic, and some examples are built with a novel interface dynamics that is not found in the literature. Also, it is shown that, on a Neumann boundary value problem, the position of the interface at thermodynamic equilibrium depends entirely on the initial temperature profile. The prediction of the interface position for large time values makes possible to fine tune the numerical methods, and given that energy conservation demands highly precise solutions, we found that it was necessary to develop a general non-classical finite difference scheme where a non-homogeneous moving mesh is considered. Numerical examples are shown to test these predictions and finally, we study the phase transition on a thermally isolated sample with a liquid and a solid phase in aluminum.