Boundary corrections for splitting methods in the time integration of multidimensional parabolic problems

TL;DR

This paper introduces boundary correction techniques for splitting methods in multidimensional parabolic PDEs, restoring the expected order of convergence under time-dependent boundary conditions using high-order ADI-type integrators.

Contribution

The paper proposes novel boundary correction techniques that enable high-order time integration methods to maintain their convergence order with time-dependent boundary conditions.

Findings

Boundary corrections restore the order of convergence for time-dependent boundary conditions.

High-order ADI-type methods achieve their theoretical order in PDE sense.

Techniques are applicable to most splitting methods with directional splitting.

Abstract

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.