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W. Auzinger, H. Hofstätter, W. Kreuzer, E. Weinmüller:
"Defect correction techniques for stiff initial value problems";
Vortrag: SciCADE 2003, Trondheim; 30.06.2003 - 04.07.2003.

In the talk given by W.Auzinger it is shown that certain modified defect correction algorithms based on techniques like defect interpolation (IPDeC) and defect quadrature (IQDeC) enable the efficient iterative realization of superconvergent collocation solutions corresponding to Gauss or Radau schemes. The excellent stability and convergence properties of these schemes for stiff ODEs motivates the application of the modified defect correction algorithms to stiff initial value problems, which is the topic of this talk.

First we present some theoretical and experimental results for the classical one-dimensional Prothero-Robinson model problem. This enables the evaluation of different algorithmic components like the one-step method to be used as the basic discretization scheme. For systems of stiff ODEs, the convergence is influenced by the degree of coupling between the stiff and the non-stiff components and by the behavior of the stiff eigendirections. For singularly perturbed problems, the Jacobian has a slowly varying stiff eigenspace, and in this case the stiff error component behaves like the error in the one-dimensional case, whereas the smooth error component shows the classical (non-stiff) behavior, featuring superconvergence over a wide range of stepsizes. This is demonstrated by means of numerical experiments. For more difficult problems with a significantly varying stiff eigendirection, however, some of the modified defect correction algorithms show an unstable behavior. We present a technique to avoid this instability, which is based on the idea of applying QR-transformation resulting in an approximate decoupling of stiff and non-stiff components.

http://publik.tuwien.ac.at/files/pub-tm_949.pdf