A comparison of boundary correction methods for Strang splitting

Abstract

In this paper we consider splitting methods in the presence of non-homogeneous boundary conditions. In particular, we consider the corrections that have been described and analyzed in Einkemmer, Ostermann 2015 and Alonso-Mallo, Cano, Reguera 2016. The latter method is extended to the non-linear case, and a rigorous convergence analysis is provided. We perform numerical simulations for diffusion-reaction, advection-reaction, and dispersion-reaction equations in order to evaluate the relative performance of these two corrections. Furthermore, we introduce an extension of both methods to obtain order three locally and evaluate under what circumstances this is beneficial.

Avoiding order reduction when integrating linear initial boundary value problems with exponential splitting methods

“…Although it is not an aim of this paper, there are already results on applying a similar technique to nonlinear problems [5,7], and [11] tries to compare with the technique in [9,10] for them.…”

Comparison of efficiency among different techniques to avoid order reduction with Strang splitting

“…We notice that, for a fixed value of k, EO2 leads to smaller errors than EO1, in the same way that happened with ACR2 with respect to ACR1. Moreover, following [17], we have also considered numerical differentiation in order to try to get local order 3 with EO1 and EO2 in (10). More precisely, theoretically, a function q should be taken for which q(t) = f (t, u(t)) and Aq(t) = Af (t, u(t)).…”

“…Moreover, following [17], we have also considered numerical differentiation in order to try to get local order 3 with EO1 and EO2 in (10). More precisely, theoretically, a function q should be taken for which ∂q ( t ) = ∂f ( t , u ( t )) and ∂Aq ( t ) = ∂Af ( t , u ( t )). Although, even when that function can be constructed, the order for the global error does not improve, it is interesting to see whether the fact that the local errors maybe smaller implies a better overall behavior.…”

“…For the numerical experiments, both 1-dimensional and bidimensional problems will be considered. There is already another comparison in the literature between both techniques [17] but there they just compare in terms of error against the time stepsize without entering into the details of implementation and its computational cost, which we believe that is the interesting comparison. Moreover, they just consider time-independent boundary conditions and 1-dimensional problems, for which many simplifications can be made.…”

Efficient boundary corrected Strang splitting

“…An application of Strang splitting to this system is called modified Strang splitting in the following presentation and has been shown to be extremely competitive compared to other boundary correction techniques [7]. We remark that this approach, in general, requires that the correction function is computed once every time step.…”