In this work, we use the lattice Boltzmann method (LBM) to simulate High Rayleigh number natural convection flows in a cavity filled with air and heated from below. One of the vertical walls has a cold portion, while the other walls are adiabatic. This cavity simulates a living space with under-floor heating and fitted with a glass door (single glass). Computations are made in two dimensions for high values of Rayleigh number. The model uses the double populations approach to simulate hydrodynamic and thermal fields. The traditional LBM with a non-uniform grid has high grid requirements at higher Rayleigh number values. The results are presented in terms of streamlines and isotherms. Variation of the local Nusselt number on the walls is shown and discussed. Global heat transfer has been studied in terms of average Nusselt number which is correlated with Rayleigh number. Dimensionless profiles of temperature and velocity along the vertical and horizontal centerline are illustrated.