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.

翻译：本文针对$d$维空间离散化抛物型问题在时变边界条件下，于矩形区域上采用两种边界修正技术，以缓解其在PDE意义下时间收敛阶的降低（即当空间与时间分辨率相互独立地趋于零时）。我们采用MoL方法（直线法），其中空间离散化采用四阶中心差分格式，时间积分则通过$s$阶段AMF-W方法实现。该时间积分器属于ADI类型（通过方向分裂实现交替方向隐式），且具有高于文献中常见方法（通常仅达二阶）的精度阶。此外，本文阐述的技术在采用方向分裂时，同样适用于大多数分裂方法。值得注意的事实是：通过这些技术，时间积分器能够在时间无关边界条件的精度水平上，恢复PDE收敛的时间阶数。