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BIT Numerical Mathematics

, Volume 57, Issue 3, pp 845–865 | Cite as

Conversion methods for improving structural analysis of differential-algebraic equation systems

  • Guangning Tan
  • Nedialko S. Nedialkov
  • John D. Pryce
Article
  • 160 Downloads

Abstract

Structural analysis (SA) of a system of differential-algebraic equations (DAEs) is used to determine its index and which equations to be differentiated and how many times. Both Pantelides’s algorithm and Pryce’s \(\varSigma \)-method are equivalent: if one of them finds correct structural information, the other does also. Nonsingularity of the Jacobian produced by SA indicates success, which occurs on many problems of interest. However, these methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates \(\varSigma \)-method’s failures and presents two conversion methods for fixing them. Under certain conditions, both methods reformulate a DAE system on which the \(\varSigma \)-method fails into a locally equivalent problem on which SA is more likely to succeed. Aiming at achieving global equivalence between the original DAE system and the converted one, we provide a rationale for choosing a conversion from the applicable ones.

Keywords

Differential-algebraic equations Structural analysis Modeling Symbolic computation 

Mathematics Subject Classification

34A09 65L80 41A58 68W30 

Notes

Acknowledgements

The authors acknowledge with thanks the financial support for this research: GT was supported in part by the McMaster Centre for Software Certification through the Ontario Research Fund, Canada, NSN was supported in part by the Natural Sciences and Engineering Research Council of Canada, and JDP was supported in part by the Leverhulme Trust, the UK. The authors thank the anonymous reviewers for providing valuable suggestions on improving this article.

References

  1. 1.
    Barrio, R.: Performance of the Taylor series method for ODEs/DAEs. Appl. Math. Comput. 163, 525–545 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Barrio, R.: Sensitivity analysis of ODEs/DAEs using the Taylor series method. SIAM J. Sci. Comput. 27, 929–1947 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 2nd edn. SIAM, Philadelphia (1996)zbMATHGoogle Scholar
  4. 4.
    Campbell, S.L., Griepentrog, E.: Solvability of general differential-algebraic equations. SIAM J. Sci. Comput. 16(2), 257–270 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carpanzano, E., Maffezzoni, C.: Symbolic manipulation techniques for model simplification in object-oriented modelling of large scale continuous systems. Math. Comput. Simul. 48(2), 133–150 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chowdhry, S., Krendl, H., Linninger, A.A.: Symbolic numeric index analysis algorithm for differential-algebraic equations. Indus. Eng. Chem. Res. 43(14), 3886–3894 (2004)CrossRefGoogle Scholar
  7. 7.
    Griewank, A.: On automatic differentiation. In: Iri, M., Tanabe, K. (eds.) Mathematical Programming: Recent Developments and Applications, pp. 83–108. Kluwer Academic Publishers, Dordrecht (1989)Google Scholar
  8. 8.
    Kunkel, P., Mehrmann, V.: Index reduction for differential-algebraic equations by minimal extension. ZAMM J. Appl. Math. Mech. Z. für Angew. Math. Mech. 84(9), 579–597 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kunkel, P., Mehrmann, V.L.: Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society, Zürich (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    MapleSim: Technological superiority in multi-domain physical modeling and simulation (2012). http://www.maplesoft.com/view.aspx?sf=7032
  11. 11.
    Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14(3), 677–692 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mazzia, F., Iavernaro, F.: Test Set for Initial Value Problem Solvers. Tech. Rep. 40, Department of Mathematics, University of Bari (2003). http://pitagora.dm.uniba.it/~testset/
  13. 13.
    Nedialkov, N., Pryce, J.D.: DAETS User Guide. Tech. Rep. CAS 08-08-NN, Department of Computing and Software, McMaster University, Hamilton (2013). pp 68, DAETS is available at http://www.cas.mcmaster.ca/~nedialk/daets
  14. 14.
    Nedialkov, N.S., Pryce, J.D.: Solving differential-algebraic equations by Taylor series (I): computing Taylor coefficients. BIT Numer. Math. 45(3), 561–591 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nedialkov, N.S., Pryce, J.D.: Solving differential-algebraic equations by Taylor series (III): the DAETS code. JNAIAM J. Numer. Anal. Indus. Appl. Math. 3, 61–80 (2008)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Nedialkov, N.S., Pryce, J.D., Tan, G.: Algorithm 948: DAESA—a Matlab tool for structural analysis of differential-algebraic equations: software. ACM Trans. Math. Softw. 41(2), 12:1–12:14 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pantelides, C.C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9, 213–231 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pryce, J.D.: Solving high-index DAEs by Taylor series. Numer. Algorithms 19, 195–211 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pryce, J.D.: A simple structural analysis method for DAEs. BIT Numer. Math. 41(2), 364–394 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pryce, J.D., Nedialkov, N.S., Tan, G.: DAESA—a Matlab tool for structural analysis of differential-algebraic equations: theory. ACM Trans. Math. Softw. 41(2), 9:1–9:20 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Richardson, D.: Some undecidable problems involving elementary functions of a real variable. J. Symb. Logic 33(4), 514–520 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Scholz, L., Steinbrecher, A.: Regularization of DAEs based on the signature method. BIT Numer. Math. 56(1), 319–340 (2016)Google Scholar
  24. 24.
    Scholz, L., Steinbrecher, A.: Structural-algebraic regularization for coupled systems of DAEs. BIT Numer. Math. 56(2), 777–804 (2016)Google Scholar
  25. 25.
    Sjölund, M., Fritzson, P.: Debugging symbolic transformations in equation systems. In: Proceedings of Equation-Based Object-Oriented Modeling Languages and Tools (EOOLT), pp. 67–74 (2011)Google Scholar
  26. 26.
    Tan, G.: Conversion methods for improving structural analysis of differential-algebraic equation systems. Ph.D. thesis, School of Computational Science and Engineering, McMaster University, 1280 Main St W, Hamilton, Ontario, L8S 1A8, Canada, pp. 168 (2016)Google Scholar
  27. 27.
    Tan, G., Nedialkov, N.S., Pryce, J.D.: Conversion methods, block triangularization, and structural analysis of differential-algebraic equation systems. Tech. Rep. CAS 16-04-NN, Department of Computing and Software, McMaster University, Hamilton, pp. 25 (2016). http://www.cas.mcmaster.ca/cas/0reports/CAS-16-04-NN.pdf
  28. 28.
    The MathWorks, Inc.: Matlab Symbolic Math Toolbox (2016). http://www.mathworks.com/products/symbolic/

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of Computational Science and EngineeringMcMaster UniversityHamiltonCanada
  2. 2.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  3. 3.School of MathematicsCardiff UniversityCardiffWales, UK

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