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IMA Journal of Numerical Analysis Advance Access originally published online on September 7, 2007
IMA Journal of Numerical Analysis 2008 28(2):274-291; doi:10.1093/imanum/drm007
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Projected Runge–Kutta methods for index 3 differential–algebraic equations near equilibria, periodic orbits and attracting sets

Johannes Schropp{dagger}

Department of Mathematics and Statistics, University of Konstanz, Universitätsstrasse 10, D-78464 Konstanz, Germany

{dagger} Email: jschropp{at}math.uni-koeln.de

Received on 23 March 2004. Revised on 15 March 2007.


  Abstract

In the present paper, we analyse the geometric properties of projected Runge–Kutta methods for the solution of index 3 differential–algebraic equations in the Hessenberg form. We show that the geometric phase portrait is well reproduced under discretization in the vicinity of equilibria, periodic orbits or asymptotically stable invariant sets. The main tools are embedding techniques and an invariant manifold theorem which allow a reduction of the problem to the classical ordinary differential equation case.

Key Words: projected Runge–Kutta method; index 3 differential–algebraic equation; geometric phase portrait; topological conjugacy


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