A general numerical method for the solution of non-steady state wave and convection dominated dispersion packed bed reactor models is presented. Several numerical problems specific for this type of equations are addressed. Special care is taken to accurately and stably discretize the convection terms. The stiffness of the model equations is tackled by a simple but robust method based on the Newton–Kantorovich procedure and accompanied with an automatic time stepping mechanism. A new technique is proposed to automatically select and solve the minimum possible number of equations. The focus of this article is on the development of simple but robust approximation techniques for dealing with particular numerical problems and their coupling in an overall algorithm. All the methods described in this article have been implemented in a software package for the mathematical modeling of catalytic packed bed reactors.