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On parallel methods for boundary value ODEs

Parallele Methoden für Randwertprobleme gewöhnlicher Differentialgleichungen

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Abstract

Some of the traditional methods for boundary value ODEs, such as standard multiple shooting, finite difference and collocation methods, lend themselves well to parallelization in the independent variable: the first stage of the construction of a solution approximation is performed independently on each subinterval of a mesh. However, the underlying possibly fast bidirectional propagation of information by fundamental modes brings about stability difficulties when information from the different subintervals is combined to form a global solution. Additional difficulties occur when a very stiff problem is to be efficiently and stably solved on a parallel architecture.

In this paper difference and parallel shooting methods are examined. A parallel algorithm for the stable solution of the resulting algebraic system is proposed and evaluated. A parallel algorithm for stiff boundary value problems is proposed as well.

Zusammenfassung

Eine REihe traditioneller Lösungsverfahren für Randwertprobleme gewöhnlicher Differentialgleichungen weisen ein natürliches Potential für deren Parallelisierung auf. Insbesondere der erste Teil der Lösungsphase, die Berechnung approximativer Lösungen auf einer Partition des Definitionsintervalles kann parallel ausgeführt werden. In der zweiten Phase, wo die lokalen Teilstücke zu einer globalen Lösung zusammengesetzt werden, treten jedoch durch die Existenz stabiler und instabiler Lösungsmoden Stabilitätsprobleme auf. Zusätzliche Probleme zeigen sich speziell dann, wenn sehr steife Randwertprobleme sowohl effizient als auch stabil auf Computern mit Parallelarchitekturen gelöst werden sollen.

Die vorliegende Arbeit untersucht Differenzenmethoden und Verfahren vom parallel Shooting Typ. Ein paralleler Algorithmus für die Lösung der auftretenden linearen Gleichungssysteme wird vorgeschlagen und analysiert. Schließlich wird ein Verfahren für steife Randwertprobleme vorgestellt.

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Authors and Affiliations

  1. Department of Computer Science, University of British Columbia, V6T 1W5, Vancouver, B. C., Canada

    U. M. Ascher & S. Y. P. Chan

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  1. U. M. Ascher
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  2. S. Y. P. Chan
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In honour of Hans J. Stetter on the occasion of his 60th birthday

This research was supported in part by NSERC Canada Grant OGP0004306.

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Ascher, U.M., Chan, S.Y.P. On parallel methods for boundary value ODEs. Computing 46, 1–17 (1991). https://doi.org/10.1007/BF02239008

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