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Convergence and comparisons of waveform relaxation methods for initial value problems of linearODE systems

Konvergenz und Vergleich der Sukzessiven Relaxation für Anfangswertprobleme linearer gewöhnlicher Differentialgleichungen

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Abstract

In this paper we investigate waveform relaxation (WR) methods for solving initial value problems of linearODE systems. Some sufficient conditions for convergence are proposed. A class ofWR AOR, WR SOR, andWR JOR methods is defined and their convergence is discussed. The asymptotic rates of convergence of two differentWR methods are compared.

Zusammenfassung

In diesem Artikel studieren wir die Sukzessive Relaxation (SR) für Anfangswert-probleme linearer gewöhnlicher Differentialgleichungen. Einige hinreichende Bedingungen für die Konvergenz werden aufgestellt. DieSR-AOR-, SR-SOR- undSR-JOR-Verfahren werden definiert und ihre Konvergenz diskutiert. Die asymptotischen Konvergenzgeschwindigkeiten von zwei verschiedenenSR-Methoden werden verglichen.

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References

  1. Berman, A., Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. New York: Academic Press 1979.

    Google Scholar 

  2. Elsner, L.: Comparisons of weak regular splittings and multisplitting methods. Numer. Math.56, 283–289 (1989).

    Google Scholar 

  3. Holmes, R. B.: A formula for the spectral radius of an operator, Amer. Math. Monthly75, 163–166 (1968).

    Google Scholar 

  4. Miekkala, U., Nevanlinna, O.: Convergence of dynamic iteration methods for initial value problems. SIAM J. Sci. Stat. Comp.8, 459–482 (1987).

    Google Scholar 

  5. Miller, V. A., Neumann, M.: A note on comparison theorems for nonnegative matrices. Numer. Math.47, 427–434 (1985).

    Google Scholar 

  6. Nevanlinna, O.: Remarks on Picard-Lindelöf iteration, Part I. BIT29, 328–346 (1989).

    Google Scholar 

  7. Nevalinna, O.: Remarks on Picard-Lindelöf iteration, Part II, BIT,29, 535–562 (1989).

    Google Scholar 

  8. Schneider, H.: Theorems onM-splittings of a singularM-matrix which depend on graph structure. Lin. Alg. Appl.58, 407–424 (1984).

    Google Scholar 

  9. Song, Y.: Comparisons of nonnegative splittings of matrices. Lin. Alg. Appl.154/156, 433–455 (1991).

    Google Scholar 

  10. Young, D. M.: Iterative solution of large linear systems. New York: Academic Press 1971.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing Normal University, 210024, Nanjing, People's Republic of China

    Y. Song

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  1. Y. Song
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Song, Y. Convergence and comparisons of waveform relaxation methods for initial value problems of linearODE systems. Computing 50, 337–352 (1993). https://doi.org/10.1007/BF02243876

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  • DOI: https://doi.org/10.1007/BF02243876

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