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Local error structures and order conditions in terms of Lie elements for exponential splitting schemes

Auzinger W., Herfort W.

Opuscula Mathematica

2014

Vol. 34, no. 2

243--256

We discuss the structure of the local error of exponential operator splitting methods. In particular, it is shown that the leading error term is a Lie element, i.e., a linear combination of higher-degree commutators of the given operators. This structural assertion can be used to formulate a simple algorithm for the automatic generation of a minimal set of polynomial equations representing the order conditions, for the general case as well as in symmetric settings.

A uniform quantitative stiff stability estimate for BDF schemes

Auzinger W., Herfort W.

Opuscula Mathematica

2006

Vol. 26, no. 2

203-227

The concepts of stability regions, A- and A(α)-stability - albeit based on scalar models - turned out to be essential for the identification of implicit methods suitable for the integration of stiff ODEs. However, for multistep methods, knowledge of the stability region provides no information on the quantitative stability behavior of the scheme. In this paper we fill this gap for the important class of Backward Differentiation Formulas (BDF). Quantitative stability bounds are derived which are uniformly valid in the stability region of the method. Our analysis is based on a study of the separation of the characteristic roots and a special similarity decomposition of the associated companion matrix.