An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs

Abstract

A computationally efficient a posteriori error estimator is introduced and analyzed for collocation solutions to linear index-1 DAEs (differential-algebraic equations) with properly stated leading term exhibiting a singularity of the first kind. The procedure is based on a modified defect correction principle, extending an established technique from the context of ordinary differential equations to the differential-algebraic case. Using recent convergence results for stiffly accurate collocation methods, we prove that the resulting error estimate is asymptotically correct. Numerical examples demonstrate the performance of this approach. To keep the presentation reasonably self-contained, some arguments from the literature on DAEs concerning the decoupling of the problem and its discretization, which is essential for our analysis, are also briefly reviewed. The appendix contains a remark about the interrelation between collocation and implicit Runge-Kutta methods for the DAE case.

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

References

  1. 1.

    Ascher, U.M., Spiteri, R.: Collocation software for boundary value differential-algebraic equations. SIAM J. Sci. Stat. Comput. 4, 938–952 (1994)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Auzinger, W., Kneisl, G., Koch, O., Weinmüller, E.: SBVP 1.0—a MATLAB solver for singular boundary value problems. Technical Report ANUM Preprint No. 2/02, Vienna University of Technology (2002)

  3. 3.

    Auzinger, W., Koch, O., Weinmüller, E.: Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31, 5–25 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Auzinger, W., Koch, O., Weinmüller, E.: Analysis of a new error estimate for collocation methods applied to singular boundary value problems. SIAM J. Numer. Anal. 42, 2366–2386 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Auzinger, W., Koch, O., Praetorius, D., Weinmüller, E.: New a posteriori error estimates for singular boundary value problems. Numer. Algorithms 40, 79–100 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Auzinger, W., Lehner, H., Weinmüller, E.: Defect-based a posteriori error estimation for index-1 DAEs. ASC Report 20/2007, Institute for Analysis and Scientific Computing, Vienna University of Technology (2007)

  7. 7.

    Balla, K., März, R.: A unified approach to linear differential algebraic equations and their adjoints. J. Anal. Appl. 21(3), 783–802 (2002)

    MATH  Google Scholar 

  8. 8.

    Campbell, S.L.: Linearization of DAE’s along trajectories. Z. Angew. Math. Phys. 46, 70–84 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Degenhardt, A.: Collocation for transferable differential-algebraic equations. Technical Report 1992-1, Humboldt University Berlin (1992)

  10. 10.

    de Hoog, F.R., Weiss, R.: Difference methods for boundary value problems with a singularity of the first kind. SIAM J. Numer. Anal. 13, 775–813 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    de Hoog, F.R., Weiss, R.: Collocation methods for singular boundary value problems. SIAM J. Numer. Anal. 15, 198–217 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Dick, A., Koch, O., März, R., Weinmüller, E.: Convergence of collocation schemes for nonlinear index 1 DAEs with a singular point, in preparation

  13. 13.

    Dokchan, R.: Numerical integration of DAEs with harmless critical points. Humboldt University Berlin, Working paper (2007)

  14. 14.

    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I— Nonstiff Problems, 2nd edn. Springer, Berlin (1993)

    Google Scholar 

  15. 15.

    Higueras, I., März, R.: Differential algebraic equations with properly stated leading term. Comput. Math. Appl. 48, 215–235 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Higueras, I., März, R., Tischendorf, C.: Stability preserving integration of index-1 DAEs. Appl. Numer. Math. 45, 175–200 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Higueras, I., März, R., Tischendorf, C.: Stability preserving integration of index-2 DAEs. Appl. Numer. Math. 45, 201–229 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Kitzler, G.: A posteriori Fehlerschätzer für Zweipunkt-Randwertprobleme mittels Defektkorrektur. Diploma Thesis, Vienna University of Technology (2010)

  19. 19.

    Koch, O., März, R., Praetorius, D., Weinmüller, E.: Collocation methods for index 1 DAEs with a singularity of the first kind. Math. Comput. 79, 281–304 (2010)

    Article  MATH  Google Scholar 

  20. 20.

    Kopelmann, A.: Ein Kollokationsverfahren für überführbare Algebro-Differentialgleichungen. Preprint 1987-151, Humboldt University Berlin (1987)

  21. 21.

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

    Google Scholar 

  22. 22.

    Kunkel, P., Stöver, R.: Symmetric collocation methods for linear differential-algebraic boundary value problems. Numer. Math. 91, 475–501 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    März, R.: Differential algebraic equations anew. Appl. Numer. Math. 42, 315–335 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    März, R., Riaza, R.: Linear index-1 DAEs: Regular and singular problems. Acta Appl. Math. 84, 24–53 (2004)

    Google Scholar 

  25. 25.

    März, R., Riaza, R.: Linear differential-algebraic equations with properly stated leading term: regular points. J. Math. Anal. Appl. 323, 1279–1299 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    März, R., Riaza, R.: Linear differential-algebraic equations with properly stated leading term: A—critical points. Math. Comput. Model. Dyn. Syst. 13, 291–314 (2007)

    MATH  MathSciNet  Google Scholar 

  27. 27.

    März, R., Riaza, R.: Linear differential-algebraic equations with properly stated leading term: B—critical points. Preprint 2007-09, Humboldt University Berlin (2007)

  28. 28.

    Riaza, R., März, R.: A simpler construction of the matrix chain defining the tractability index of linear DAEs. Appl. Math. Lett. 21(4), 326–331 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. 29.

    Saboor Bagherzadeh, A.: Defect-based error estimation for higher order differential equations. PhD Thesis, Vienna University of Technology (2011, in preparation)

  30. 30.

    Schulz, S.: Four Lectures on Differential-Algebraic Equations. Report Series 497, Dept. of Mathematics, Univ. of Auckland (2003)

  31. 31.

    Stetter, H.J.: The defect correction principle and discretization methods. Numer. Math. 29, 425–443 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  32. 32.

    Zadunaisky, P.E.: On the estimation of errors propagated in the numerical integration of ODEs. Numer. Math. 27, 21–39 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  33. 33.

    Zielke, G.: Motivation und Darstellung von verallgemeinerten Matrixinversen. Beitr. Numer. Math. 7, 177–218 (1979)

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Winfried Auzinger.

Additional information

Communicated by Christian Lubich.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Auzinger, W., Lehner, H. & Weinmüller, E. An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs. Bit Numer Math 51, 43–65 (2011). https://doi.org/10.1007/s10543-011-0321-9

Download citation

Keywords

  • Differential algebraic equations
  • Singularity of the first kind
  • Collocation
  • A posteriori error estimation
  • Defect correction

Mathematics Subject Classification (2000)

  • 65L80
  • 65B05