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.

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