For the numerical solution of parabolic problems with linear diffusion term, linearly implicit time integrators are considered. To reduce the cost on the linear algebra level an alternating direction implicit (ADI) approach is applied (so-called AMF-W-methods). The present work proves optimal bounds of the global error for two classes of 1-stage methods in the Euclidean 2 norm as well as in the maximum norm ∞. The bounds are valid under a very weak step size restriction that covers PDE-convergence, where the time step size is of the same order as the spatial grid size.
