Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2006
Volume
Vol. 26
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 26 (2006)
Projects
Pages
Articles
A uniform quantitative stiff stability estimate for BDF schemes
(2006) Auzinger, Winfried; Herfort, Wolfgang
The concepts of stability regions, $A$- and $A(\alpha)$-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.
Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations
(2006) Auzinger, Winfried; Karner, Ernst; Koch, Othmar; Weinmüller, Ewa
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.
Application of Green's operator to quadratic variational problems
(2006) Azbelev, Nikolay V.; Tsalyuk, Vadim Z.
We use Green’s function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space $L_{2}$ of quadratically summable functions. We get the neccessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert–Schmidt operator in $L_{2}$ with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1) a variational problem with deviating argument, and 2) the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar’s axis).
On reconstruction of structure of a linear system with time delay
(2006) Blizorukova, Marina; Fedina, Nina; Maksimov, Vyacheslav
The problem of reconstruction of a structure of a linear system with delay is considered. A solution algorithm stable with respect to the informational noise and computational errors is specified. The algorithm is based on the method of auxiliary positionally controlled models.
The numerical solution of nonlinear two-point boundary value problems using iterated deferred correction - a survey
(Wydawnictwa AGH, 2006) Cash, Jeff R.
The use of iterated deferred correction has proved to be a very efficient approach to the numerical solution of general first order systems of nonlinear two-point boundary value problems. In particular the two high order codes TWPBVP.f, based on mono-implicit Runge–Kutta (MIRK) formulae, and TWPBVPL.f based on Lobatto Runge–Kutta formulae as well as the continuation codes ACDC.f and COLMOD.f are now widely used. In this survey we describe some of the problems involved in the derivation of efficient deferred correction schemes. In particular we consider the construction of high order methods which preserve the stability of the underlying formulae, the choice of the mesh choosing algorithm which is based both on local accuracy and conditioning, and the computation of continuous solutions.