A high accuracy numerical model is used to simulate an alternate melting and solidification cycle of a phase change material (PCM). We use a second order (in time and space) finite-element method with mesh adaptivity to solve a single-domain model based on the Navier-Stokes-Boussinesq equations. An enthalpy method is applied to the energy equation. A Carman-Kozeny type penalty term is introduced in the momentum equation to bring the velocity to zero inside the solid region. The mesh is dynamically adapted at each time step to accurately capture the interface between solid and liquid phases, the boundary-layer structure at the walls and the multi-cellular unsteady convection in the liquid. We consider the basic configuration of a differentially heated square cavity filled with an octadecane paraffin and use experimental and numerical results from the literature to validate our numerical system. The first study case considers the complete melting of the PCM (liquid fraction of 95%), followed by a complete solidification. For the second case, the solidification is triggered after a partial melting (liquid fraction of 50%). Both cases are analysed in detail by providing temporal evolution of the solid- liquid interface, liquid fraction, Nusselt number and accumulated heat input. Different regimes are identified during the melting-solidification process and explained using scaling correlation analysis. Practical consequences of these two operating modes are finally discussed.