Grid refinement has been addressed by different authors in the lattice Boltzmann method community. The information communication and reconstruction on grid transitions is of crucial importance from the accuracy and numerical stability point of view. While a decimation is performed when going from the fine to the coarse grid, a reconstruction must performed when going form the coarse to the fine grid. In this paper we analyze these two steps. We first show that for the decimation operation, a simple copy of the information from the fine to the coarse grid is not sufficient to guarantee the stability of the numerical scheme at high Reynolds numbers, but that a filtering operation must be added. Then we demonstrate that to reconstruct the information, a local cubic interpolation scheme is mandatory in order to get a precision compatible with the order of accuracy of the lattice Boltzmann method. These two fundamental extra-steps are validated on two classical 2D bench- marks, the 2D circular cylinder and the 2D dipole-wall collision. The latter is especially challenging from the numerical point of view since we allow strong gradients to cross the refinement interfaces at a relatively high Reynolds number of 5000. A very good agreement is found between the single grid and the refined grid cases. The proposed grid refinement strategy has been implemented in the parallel open-source library Palabos.