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Detecting and locating a singular point in the numerical solution of IVPs for ODEs

Die Erkennung und Lokalisierung einer singulären Stelle bei der numerischen Lösung von AWPen für gewöhnliche Differentialgleichungen

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Abstract

Many authors have worked on approaches for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) whose solutions contain one or more singular points within the interval of integration. Their approaches, however, assumed that the user knows in advance that the problem is singular. Hence they introduced new formulas to cope with this difficulty. In this paper, a new approach to detect and locate a singularity is suggested. This approach, which does not require the changing of the underlying formula, is comprised of two stages. The first is a preliminary singularity detection stage. The second stage is the confirmation stage which gathers more information about the existence and location of the singular point. We justify the first state and introduce three different techniques for confirming the existence of a singularity. The numerical results show that our approach is effective.

Zusammenfassung

Es gibt eine Reihe von Ansätzen zur numerischen Behandlung von Anfangswertproblemen für gewöhnliche Differentialgleichungen, deren Lösungen eine oder mehrere singuläre Stellen im Integrationsintervall enthalten. Diese Ansätze nehmen jedoch an, daß der Benutzer im voraus weiß, daß das Problem singulär ist; es werden deshalb neue Formeln eingeführt, die auf diese Schwierigkeiten zugeschnitten sind. In unserer Arbeit wird ein neuer Zugang zur Erkennung und Lokalisierung einer Singularität vorgeschlagen. Er erfordert keine Veränderung der zugrundeliegenden Integrationsformel und besteht aus zwei Stufen. Die erste Stufe dient zum Hinweis auf eine vermutete Singularität. Die zweite Stufe, die mehr Information bezüglich der Existenz und Lage der Singularität aufbaut, dient der Bestätigung des Vorhandenseins. Wir begründen unsere erste Stufe und geben drei verschiedene Techniken zur Bestätigung des Vorhandenseins an. Die numerischen Ergebnisse zeigen die Wirksamkeit unseres Vorgehens.

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Authors and Affiliations

  1. Computer Science Center, University of Indonesia, Tl Salemba 4, P.O.B. 3442, 10002, Jakarta, Indonesia

    Heru Suhartanto

  2. Department of Computer Science, University of Toronto, M5S 1A4, Canada

    Wayne H. Enright

  3. U.S.A.

    Wayne H. Enright

Authors
  1. Heru Suhartanto
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  2. Wayne H. Enright
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Supported by the Information Technology Research Centre of Ontario, and the Natural Science and Engineering Research Council of Canada.

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Suhartanto, H., Enright, W.H. Detecting and locating a singular point in the numerical solution of IVPs for ODEs. Computing 48, 161–175 (1992). https://doi.org/10.1007/BF02310531

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  • DOI: https://doi.org/10.1007/BF02310531

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