We employ numerical strategies to investigate the diffusive-thermal instability of adiabatic reverse smolder waves and the subsequent evolution processes of the smolder waves following the onset of instability. First, the linear stability of adiabatic reverse smolder waves is assessed using the numerical normal mode analysis method. This analysis generates a stability diagram characterized by the oxygen Lewis number and the incoming air flow rate M_g. Two different types of instabilities are identified: cellular (or fingering) instability and traveling wave instability. The categorization depends on the range of the control parameters. The linear stability analysis results are validated through a comparison with numerical simulations of 2-D nonlinear evolution processes of smolder waves.