A numerical algorithm for the solution of systems of parabolic partial differential equations in one space dimension with a fully adaptive time and space grid control is presented. The adaptivity is based upon a detailed estimation of the errors coming from the discretizations of both space and time. The basic features of the error estimation procedure and its implementation into the code PDEX are discussed. The advantages of the method are shown through comparison with conventional algorithms for simulations of reversed flow fixed bed reactors and the startup behaviour of automobile catalytic converters.