A comparison of boundary correction methods for Strang splitting

Lukas Einkemmer,,Department Of Mathematics, University Of Innsbruck, A-6020 Innsbruck, Austria ,Alexander Ostermann

doi:10.3934/dcdsb.2018081

Lukas Einkemmer, ,Department Of Mathematics, University Of Innsbruck, A-6020 Innsbruck, Austria + Show 1 more

Open Access

https://doi.org/10.3934/dcdsb.2018081

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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.

Highlights

The present paper is concerned with the numerical solution of the partial differential equations that can be written as an abstract evolution equation
In the present work we will consider in particular A = ∂xx, A = ∂x(a(x) · ), and A = i∂xx
Let us emphasize that dependent on the problem type, boundary conditions can only be prescribed at a certain part of the boundary

Summary

Introduction

Where f (u) is a non-stiff reaction (usually f does only depend on u but not on its derivatives) and A is a linear differential operator. The latter is the reason for the stiffness of the spatial semi-discretization. In the part of the analysis common to all problem classes introduced above, we will use ∂Ω to denote the part of the boundary at which conditions are imposed. This is mainly done for notational simplicity

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