Cyclic adsorption processes attain a cyclic-steady state (CSS) condition by undergoing repeated cycles in time, owing to their transient and modular nature. Mathematically, solving a set of underlying nonlinear partial differential equations iteratively for different steps in a cycle until the CSS condition is attained presents a computational challenge, making the simulation and optimization of cyclic adsorption processes time-consuming. This paper focuses on expediting the CSS convergence in adsorption process simulations by implementing two vector-based acceleration methods that offer quadratic convergence akin to Newton’s methods. These methods are straightforward to implement, requiring no prior knowledge of the first derivatives (or Jacobian). The study demonstrates the efficacy of accelerated convergence by considering two adsorption processes that exhibit complex dynamics, namely, a four-step vacuum swing adsorption and a six-step temperature swing adsorption cycles for post-combustion CO2 capture. The case studies showcase the potential for improved computational efficiency in adsorption process simulations.