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Diagnosis of singular points of structured DAEs using automatic differentiation

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Abstract

At a singular point of a DAE, the IVP fails to have a unique solution. Hence, numerical integration methods cannot provide reasonable results. Unfortunately, common error control strategies do not always detect these circumstances and arbitrary solutions may be given to the user without warnings of any kind.

Automatic (or Algorithmic) Differentiation (AD) opens new possibilities to realize an analysis of DAEs and to monitor assumptions required for the existence and uniqueness of IVPs. We show how the diagnosis of singular points can be performed for structured quasi-linear DAEs up to index 2. Our approach uses the projector based analysis for DAEs employing AD. The resulting method is illustrated by several examples, with particular emphasis on simple electrical circuits containing controlled sources.

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References

  1. Estévez Schwarz, D.: Consistent initialization for index-2 differential algebraic equations and its application to circuit simulation. Humboldt-Univ., Mathematisch-Naturwissenschaftliche Fakultaẗ II. PhD thesis, Berlin (2000). http://edoc.hu-berlin.de/docviews/abstract.php?id=10218

  2. Estévez Schwarz, D., Lamour, R.: Monitoring singularities while integrating DAEs. Differential-Algebraic Equations Forum. Springer (To appear)

  3. Estévez Schwarz D., Lamour R.: Projector based integration of DAEs with the Taylor series method using automatic differentiation. J. Comput. Appl. Math. (2014)

  4. Estévez Schwarz, D., Tischendorf, C.: Structural analysis of electric circuits and consequences for the MNA. Int. J. Circuit Theory Appl. 28(2), 131–162 (2000)

    Article  MATH  Google Scholar 

  5. Golub, G.H., van Loan, C.F.: Matrix Computations. John Hopkins University Press, Baltimore and London (1996)

    MATH  Google Scholar 

  6. Griepentrog, E., März, R.: Differential-algebraic equations and their numerical treatment., volume 88 of Teubner-Texte zur Mathematik. B.G. Teubner Verlagsgesellschaft, Leipzig (1986)

  7. Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Springer (1993)

  8. Lamour, R., März, R., Tischendorf, C.: Differential-algebraic equations: A projector based analysis. Differential-Algebraic Equations Forum, vol. 1. Springer, Berlin (2013)

    Book  Google Scholar 

  9. Mazzia, F., Magherini, C.: Test set for initial value problems release 2 Technical report, Department of Mathematics, University of Bari and INdAM, vol. 4, Research Unit of Bari (2008)

  10. Rabier, P.J., Rheinboldt, W.C.: On impasse points of quasilinear differential-algebraic equations. J. Math. Anal. Appl. 181(2), 429–454 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rabier, P.J., Rheinboldt, W.C.: On the computation of impasse points of quasi-linear differential- algebraic equations. Math. Comput. 62(205), 133–154 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Riaza, R.: Differential-algebraic systems. Analytical aspects and circuit applications. Hackensack. World Scientific, NJ (2008)

    Book  Google Scholar 

  13. Rump. S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp 77–104. SCAN-98, Kluwer Academic Publishers (1999)

  14. Tuomela, J.: On singular points of quasilinear differential and differential-algebra equations. BIT 37(4), 968–977 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Beuth Hochschule für Technik Berlin, Berlin, Germany

    Diana Estévez Schwarz

  2. Humboldt-Universität zu Berlin, Berlin, Germany

    René Lamour

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  1. Diana Estévez Schwarz
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  2. René Lamour
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Correspondence to Diana Estévez Schwarz.

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Estévez Schwarz, D., Lamour, R. Diagnosis of singular points of structured DAEs using automatic differentiation. Numer Algor 69, 667–691 (2015). https://doi.org/10.1007/s11075-014-9919-8

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  • DOI: https://doi.org/10.1007/s11075-014-9919-8

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