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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.

Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations

Auzinger W., Karner E., Koch O., Weinmuller E.

Opuscula Mathematica

2006

Vol. 26, no. 2

229-241

We demonstrate that eigenvalue problems for ordinary differential equations can be recast in a formulation suitable for the solution by polynomial collocation. It is shown that the well-posedness of the two formulations is equivalent in the regular as well as in the singular case. Thus, a collocation code equipped with asymptotically correct error estimation and adaptive mesh selection can be successfully applied to compute the eigenvalues and eigenfunctions efficiently and with reliable control of the accuracy. Numerical examples illustrate this claim.

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.