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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Collocation methods for index 1 DAEs with a singularity of the first kind


Authors: Othmar Koch, Roswitha März, Dirk Praetorius and Ewa Weinmüller
Journal: Math. Comp. 79 (2010), 281-304
MSC (2000): Primary 65L80; Secondary 65L70
Published electronically: June 25, 2009
MathSciNet review: 2552227
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Abstract: We study the convergence behavior of collocation schemes applied to approximate solutions of BVPs in linear index 1 DAEs which exhibit a critical point at the left boundary. Such a critical point of the DAE causes a singularity within the inherent ODE system. We focus our attention on the case when the inherent ODE system is singular with a singularity of the first kind, apply polynomial collocation to the original DAE system and consider different choices of the collocation points such as equidistant, Gaussian or Radau points. We show that for a well-posed boundary value problem for DAEs having a sufficiently smooth solution, the global error of the collocation scheme converges with the order $ O(h^s)$, where $ s$ is the number of collocation points. Superconvergence cannot be expected in general due to the singularity, not even for the differential components of the solution. The theoretical results are illustrated by numerical experiments.


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Additional Information

Othmar Koch
Affiliation: Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8–10, A-1040 Wien, Austria
Email: othmar@othmar-koch.org

Roswitha März
Affiliation: Humboldt-Universität of Berlin, Institute for Mathematics, Unter den Linden 6, D-10099 Berlin, Germany
Email: maerz@mathematik.hu-berlin.de

Dirk Praetorius
Affiliation: Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8–10, A-1040 Wien, Austria
Email: dirk.praetorius@tuwien.ac.at

Ewa Weinmüller
Affiliation: Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8–10, A-1040 Wien, Austria
Email: ewa.weinmueller@tuwien.ac.at

DOI: http://dx.doi.org/10.1090/S0025-5718-09-02267-4
PII: S 0025-5718(09)02267-4
Received by editor(s): April 29, 2008
Received by editor(s) in revised form: March 4, 2009
Published electronically: June 25, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.