Accurate numerical solution of cyclic adsorption problems can be challenging and time consuming due to the inherent non-linearity of the system conservation equations and the presence of strong temporal and spatial gradients. In addition, the wide Variety of possible boundary conditions makes solution of a general cycle problematic. In this study we present a quadratic upwind finite volume technique that is at least second-order accurate in the spatial dimension and can easily handle any type of boundary condition (both one and two point) at either end of the adsorption column. The effect of bed pressure drop is readily accommodated. The oscillations at sharp gradients common in high-order upwind methods is significantly reduced by a solution-adaptive strategy which does not introduce significant diffusion unlike first order upwind methods. To ensure accurate: simulation of sharp fronts, we find that the finite volume node width should be approximately one tenth of the width of the mass transfer zone. A comparison is made for linear isotherm systems with the analytic solution and excellent agreement is produced for very moderate grid sizing. The method is fast and robust and is ideally suited to cycle simulation.