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Monitoring Singularities While Integrating DAEs

Conference paper
Part of the Differential-Algebraic Equations Forum book series (DAEF)

Abstract

Modern simulation tools for ODEs/DAEs allow a direct input of equations that are solved at the push of a button. However, if the mathematical assumptions that guarantee the correctness of the solution are not given, then no reliable results can be expected. Automatic (or algorithmic) differentiation (AD) opens new possibilities to analyze and solve ODEs/DAEs. In this paper, we outline how the index determination, the computation of consistent initial values, the integration and the diagnosis of singular points can be reliably carried out for DAEs up to index 3. The approach uses the projector based analysis for DAEs employing AD.

Keywords

Differential-algebraic equations Taylor series Singularity Automatic differentiation 

Notes

Acknowledgements

The authors are indebted to Roswitha März for many fruitful discussions.

References

  1. 1.
    Dokchan, R.: Numerical intergration of differential-algebraic equations with harmless critical points. PhD thesis, Humboldt-University of Berlin (2011)Google Scholar
  2. 2.
    England, R., Gómez, S., Lamour, R.: Expressing optimal control problems as differential algebraic equations. Comput. Chem. Eng. 29(8), 1720–1730 (2005)CrossRefGoogle Scholar
  3. 3.
    England, R., Gómez, S., Lamour, R.: The properties of differential-algebraic equations representing optimal control problems. Appl. Numer. Math. 59(10), 2357–2373 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Estévez Schwarz, D., Lamour, R.: Projector based integration of DAEs with the Taylor series method using automatic differentiation. J. Comput. Appl. Math. 262, 62–72 (2014)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Jackson, R.: Optimal use of mixed catalyst for two successive chemical reactions. J. Optim. Theory Appl. 2(1), 27–39 (1968)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum 1. Springer, Berlin (2013)Google Scholar
  7. 7.
    März, R., Riaza, R.: Linear index-1 DAEs: regular and singular problems. Acta Appl. Math. 84, 29–53 (2004)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    März, R., Riaza, R.: Linear differential-algebraic equations with properly stated leading term: a-critical points. Math. Comput. Model. Dyn. Syst. 13(3), 291–314 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Reissig, G., Martinson, W.S., Barton, P.I.: Differential-algebraic equations of index 1 may have an arbitrarily high structural index. SIAM J. Sci. Comput. 21(6), 1987–1990 (2000)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Riaza, R.: Differential-Algebraic Systems. Analytical Aspects and Circuit Applications. World Scientific, Hackensack (2008)CrossRefMATHGoogle Scholar
  11. 11.
    Rump, S.M.: INTLAB – INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, SCAN-98, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.B. Hochschule für Technik BerlinBerlinGermany
  2. 2.Institut für MathematikHumboldt Universität zu BerlinBerlinGermany

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