Solidification and fusion belong to a class of transient heat transfer problems known as phase change problems or moving boundary problems. The solution of this class of problems presents an additional difficulty concerning the movement of the interface. This movement is due to the absorption or removal of the latent heat at the interface. As a consequence the position of the interface is not known, being part of the solution. At the present work, the transient phase change of a pure substance is considered in the presence of natural convection in a closed two dimensional cavity. The interface is a well-defined boundary at the phase change temperature. The liquid phase is assumed to be Newtonian and the Boussinesq approximation is adopted. The properties of both liquid and solid phases are constant, although different of each other. A non-orthogonal coordinate system, which adapts to the geometry, is employed. This coordinate system moves with time to adapt to the varying interface position. The intensity of the fluid movement promotes changes in the interface shape, and it is extremely important for the phase change phenomena. At the beginning of the phase change process, the heat transfer mechanism at the liquid phase is due only to conduction. As the fluid velocity increases, the heat transfer by convection begins to dominate the process. The flow is laminar, and eventually the fluid flow becomes turbulent, substantially increasing the heat transfer rate along the interface. Further, since the fluid particles move more rapidly, theses heat fluxes along the interface are better distributed, causing a reduction of the interface curvature. The turbulence model selected belongs to the k-e family. The traditional k-e é employed at the turbulent core and another set of equations, developed based on direct numerical simulation data, is employed at the near wall region. The methodology is capable of determining the transition from laminar to turbulent flow. The present works presents a new methodology to determine the interface between solid and liquid regions. A zero thickness control volume represents the interface position. Once the mass and energy balance equations are solved at the interface, no further schemeis necessary to evaluate its new position. The zero thickness control volume at the interface allows the mass to be conserved at the liquid region without the need of any special treatment, in spite of the specific mass jump across the interface. The grid distribution is adjusted between the liquid and solid phase during the phase change process, in order to optimize the grid distribution in the domain. Further, the grid redistribution allows the use of larger time steps, without convergence difficulties. The numerical results are compared with experimental and numerical data available in the literature for fusion and solidification of pure substances. The good agreement reveals that the presented methodology furnishes an improved solution for this type of problems. The point redistribution allows the specification of larger time steps without compromising the convergence and precision.