Hydrodynamic stability during advection combined with diffusion and adsorption of a binary gas mixture in porous media is considered in this work. The general framework is that of pressure swing adsorption (PSA) on beds of zeolite beads. The dispersion/adsorption problem is studied at Darcy's scale on a 1D porous column from a numerical point of view while complete time-periodic cycles are considered. The regular solution of the problem is first sought using a finite volume scheme and stability analysis is performed on this time-dependent solution. While instability evidence is shown on a direct 2D numerical simulation of the problem with pressure and concentration disturbances at the entrance of the column, the stability analysis is carried out in 1D by making use of the linear amplitude equation method. Results clearly show the stability conditions which are quite different depending on whether pressure or concentration disturbances are considered as initial conditions or maintained as boundary conditions at column entrance.