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BIT Numerical Mathematics

, Volume 51, Issue 1, pp 43–65 | Cite as

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

  • Winfried Auzinger
  • Herbert Lehner
  • Ewa Weinmüller
Article
  • 79 Downloads

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.

Keywords

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

Mathematics Subject Classification (2000)

65L80 65B05 

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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) CrossRefMathSciNetGoogle 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) Google Scholar
  3. 3.
    Auzinger, W., Koch, O., Weinmüller, E.: Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31, 5–25 (2002) CrossRefzbMATHMathSciNetGoogle 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) CrossRefzbMATHMathSciNetGoogle 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) CrossRefzbMATHMathSciNetGoogle 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) Google Scholar
  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) zbMATHGoogle Scholar
  8. 8.
    Campbell, S.L.: Linearization of DAE’s along trajectories. Z. Angew. Math. Phys. 46, 70–84 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Degenhardt, A.: Collocation for transferable differential-algebraic equations. Technical Report 1992-1, Humboldt University Berlin (1992) Google Scholar
  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) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    de Hoog, F.R., Weiss, R.: Collocation methods for singular boundary value problems. SIAM J. Numer. Anal. 15, 198–217 (1978) CrossRefzbMATHMathSciNetGoogle 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 Google Scholar
  13. 13.
    Dokchan, R.: Numerical integration of DAEs with harmless critical points. Humboldt University Berlin, Working paper (2007) Google Scholar
  14. 14.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I— Nonstiff Problems, 2nd edn. Springer, Berlin (1993) zbMATHGoogle Scholar
  15. 15.
    Higueras, I., März, R.: Differential algebraic equations with properly stated leading term. Comput. Math. Appl. 48, 215–235 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Higueras, I., März, R., Tischendorf, C.: Stability preserving integration of index-1 DAEs. Appl. Numer. Math. 45, 175–200 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Higueras, I., März, R., Tischendorf, C.: Stability preserving integration of index-2 DAEs. Appl. Numer. Math. 45, 201–229 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kitzler, G.: A posteriori Fehlerschätzer für Zweipunkt-Randwertprobleme mittels Defektkorrektur. Diploma Thesis, Vienna University of Technology (2010) Google Scholar
  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) CrossRefzbMATHGoogle Scholar
  20. 20.
    Kopelmann, A.: Ein Kollokationsverfahren für überführbare Algebro-Differentialgleichungen. Preprint 1987-151, Humboldt University Berlin (1987) Google Scholar
  21. 21.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations—Analysis and Numerical Solution. EMS Publishing House, Zurich (2006) CrossRefzbMATHGoogle Scholar
  22. 22.
    Kunkel, P., Stöver, R.: Symmetric collocation methods for linear differential-algebraic boundary value problems. Numer. Math. 91, 475–501 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    März, R.: Differential algebraic equations anew. Appl. Numer. Math. 42, 315–335 (2002) CrossRefzbMATHMathSciNetGoogle 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) CrossRefzbMATHMathSciNetGoogle 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) zbMATHMathSciNetGoogle 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) Google Scholar
  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) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Saboor Bagherzadeh, A.: Defect-based error estimation for higher order differential equations. PhD Thesis, Vienna University of Technology (2011, in preparation) Google Scholar
  30. 30.
    Schulz, S.: Four Lectures on Differential-Algebraic Equations. Report Series 497, Dept. of Mathematics, Univ. of Auckland (2003) Google Scholar
  31. 31.
    Stetter, H.J.: The defect correction principle and discretization methods. Numer. Math. 29, 425–443 (1978) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Zadunaisky, P.E.: On the estimation of errors propagated in the numerical integration of ODEs. Numer. Math. 27, 21–39 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Zielke, G.: Motivation und Darstellung von verallgemeinerten Matrixinversen. Beitr. Numer. Math. 7, 177–218 (1979) MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  • Winfried Auzinger
    • 1
  • Herbert Lehner
    • 1
  • Ewa Weinmüller
    • 1
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria

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