Skip to main content
SpringerLink
Log in

A new projector based decoupling of linear DAEs for monitoring singularities

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

For higher index differential-algebraic equations (DAEs) some components of the solution depend on derivatives of the right-hand side. In this context, two main results are pointed out here. On the one hand, a description of the different types of undifferentiated components involved in the DAE is obtained by a projector-based decoupling. To this end, we define a new decoupling based on the number of inherent differentiations of the right-hand side that are required to determine each component. On the other hand, we introduce characteristic values that characterize the robustness of our numerically determined index-classification and decoupling as well as a meaningful indicator that permit the diagnosis of singular points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berger, T., Ilchmann, A.: On the standard canonical form of time-varying linear DAEs. Q. Appl. Math. 71(1), 69–87 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Unabridged, Corr. Republ. Classics in Applied Mathematics, vol. 14. SIAM, Society for Industrial and Applied Mathematics, Philadelphia (1996)

  3. Campbell, S.L.: A general form for solvable linear time varying singular systems of differential equations. SIAM J. Math. Anal. 18(4), 1101–1115 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Estévez Schwarz, D., da Fonseca, C.M.: On singular values related to DAEs in Kronecker canonical form. Mediterr. J. Math. (2015). doi:10.1007/s00009-015-0657-5

    Google Scholar 

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

    MATH  Google Scholar 

  6. Estévez Schwarz, D., Lamour, R.: Diagnosis of singular points of structured DAEs using automatic differentiation Numerical Algorithms 69(4), 667–691 (2015)

  7. Estévez Schwarz, D., Lamour, R.: Diagnosis of singular points of properly stated DAEs using automatic differentiation Numerical Algorithms 70(4), 777–805 (2015)

  8. 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)

  9. Galántai, A.: Projectors and Projection Methods Advances in Mathematics (Dordrecht), vol. 6. Kluwer Academic Publishers, Boston (2004)

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations - Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)

    Book  MATH  Google Scholar 

  13. Lamour, R., März, R., Tischendorf, C.: Differential-algebraic equations: A Projector Based Analysis. Differential-Algebraic Equations Forum, vol. 1. Springer, Berlin (2013)

  14. Rabier, P.J., Rheinboldt. W.C.: Theoretical and Numerical analysis of differential-algebraic equations. ciarlet, P. G. (ed.) Handbook of numerical analysis. Vol. 8: Techniques of scientific computing (Part 4). Numerical methods of fluids (Part 2), pp 183–540. Elsevier, Amsterdam (2002)

  15. Riaza, R.: Differential-algebraic systems. Analytical aspects and circuit applications. World scientific, Hackensack (2008)

  16. Walter, S.F., Lehmann, L.: Algorithmic differentiation in python with algopy. Journal of Computational Science 4(5), 334–344 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Beuth Hochschule für Technik Berlin, Berlin, Germany

    Diana Estévez Schwarz

  2. Humboldt-University of Berlin, Berlin, Germany

    René Lamour

Authors
  1. Diana Estévez Schwarz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. René Lamour
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diana Estévez Schwarz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schwarz, D.E., Lamour, R. A new projector based decoupling of linear DAEs for monitoring singularities. Numer Algor 73, 535–565 (2016). https://doi.org/10.1007/s11075-016-0107-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0107-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation