A comparison of boundary correction methods for Strang splitting

TL;DR

This paper compares boundary correction methods for Strang splitting in PDEs, extending one method to non-linear cases, providing convergence analysis, and evaluating their performance through numerical simulations.

Contribution

It extends a boundary correction method to non-linear problems, offers a rigorous convergence analysis, and compares their effectiveness in various PDE scenarios.

Findings

The extended method achieves third-order accuracy locally.

Numerical results show differences in performance depending on the problem.

The study identifies conditions where higher-order corrections are beneficial.

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.