  This work considers two boundary correction techniques to mitigate the
reduction in the temporal order of convergence in PDE sense (i.e., when both
the space and time resolutions tend to zero independently of each other) of $d$
dimension space-discretized parabolic problems on a rectangular domain subject
to time dependent boundary conditions. We make use of the MoL approach (method
of lines) where the space discretization is made with central differences of
order four and the time integration is carried out with $s$-stage
AMF-W-methods. The time integrators are of ADI-type (alternating direction
implicit by using a directional splitting) and of higher order than the usual
ones appearing in the literature which only reach order 2. Besides, the
techniques here explained also work for most of splitting methods, when
directional splitting is used. A remarkable fact is that with these techniques,
the time integrators recover the temporal order of PDE-convergence at the level
of time-independent boundary conditions.
