The use of phase-change material to enhance the capacity of energy storage/release is the subject of many new researches on management of the energy supply. Study of these systems is directly related to the solid-liquid phase-change problem, in which the evaluation of temperature distribution, position of phase-change front and liquid or solid fraction becomes a basic problem. Study of freezing and melting process with regard to natural convection in the liquid phase is the main purpose of the present paper. For this purpose, a rectangular finned container of phase-change material is intended. Fins are used to enhance the heat transfer rate. This fact necessitates the use of immersed boundary condition on the solid phase. Hence, the melting process considering the both effects of natural convection and movement of solid phase is studied. The freezing process is also studied taking into accounts the natural convection with no need to impose the immersed boundary condition. Lattice Boltzmann method is used as a numerical method and results are reported based on the dimensionless parameters. Based on the results, the effects of natural convection is negligible during freezing process, while imposing the effects of natural convection provides a significant change in the required time for complete melting of the phase change material.