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Numerical integration of differential algebraic systems and invariant manifolds

by K. Nipp

(Report number 1999-12)

Abstract
The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated in [6] for Runge-Kutta methods and in [7] for linear multistep methods. As main result, it is shown that also for Runge-Kutta methods and linear multistep methods applied to an index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.

Keywords:

@Techreport{N99_247,
  author = {K. Nipp},
  title = {Numerical integration of differential algebraic systems and invariant manifolds},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1999-12},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1999/1999-12.pdf },
  year = {1999}
}

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