An algorithm based on rigorous mathematical analysis is presented for determining the cyclic steady-state (CSS) in the simulation of Pressure Swing Adsorption (PSA) processes. The algorithm is founded on the observation that cyclic simulation of PSA processes is a Fixed-Point Iteration problem whose fixed point is the CSS. Therefore, instead of reducing the difference between the bed profiles of some dependent variables in two successive simulated cycles below a chosen tolerance value, the proposed algorithm reduces the difference between their current states and the CSS values to a set tolerance without requiring any a priori knowledge of the CSS. Three assumptions are made (1) convergence is linear in some neighbourhood of the CSS solution (2) the axial profiles at the end of the high pressure adsorption step moves only in one direction with respect to cycle number (3) the initial and ultimate convergence rates do not differ markedly. We demonstrate the effectiveness of the algorithm by applying it to a number of PSA/VSA/TSA processes simulated in the literature. For all the cases investigated, approach to the true CSS within a stated tolerance is achieved as indicated by the algorithm. (C) 2016 Elsevier Ltd. All rights reserved.