The present paper extends the bounding operator approach developed for boundary value problems to the case of initial-boundary value problems (IBVPs). Following the general principle of bounding operators enclosing methods for the case of partial differential equations are discussed. In particular, continuous discretization methods with an appropriate error bound controlled shift and monotone extensions of Rothe's method for parabolic problems are investigated. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.