Stable Numerical Solution for a Class of Structured Differential-Algebraic Equations by Linear Multistep Methods

Abstract

It is known that when we apply a linear multistep method to differential-algebraic equations (DAEs), usually the strict stability of the second characteristic polynomial is required for the zero stability. In this paper, we revisit the use of linear multistep discretizations for a class of structured strangeness-free DAEs. Both explicit and implicit linear multistep schemes can be used as underlying methods. When being applied to an appropriately reformulated form of the DAEs, the methods have the same convergent order and the same stability property as applied to ordinary differential equations (ODEs). In addition, the strict stability of the second characteristic polynomial is no longer required. In particular, for a class of semi-linear DAEs, if the underlying linear multistep method is explicit, then the computational cost may be significantly reduced. Numerical experiments are given to confirm the advantages of the new discretization schemes.

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

References

  1. 1.

    Ascher, U., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)

    Google Scholar 

  2. 2.

    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. SIAM, Philadelphia (1996)

    Google Scholar 

  3. 3.

    Bulatov, M.V., Linh, V.H., Solovarova, L.S.: On BDF-based multistep schemes for some classes of linear differential-algebraic equations of index at most 2. Acta Math. Vietnam. 41(4), 715–730 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Dokchan, R.: Numerical Integration of Differential-Algebraic Equations with Harmless Critical Point. Ph.D. thesis Humboldt-University of Berlin (2011)

  5. 5.

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

    Google Scholar 

  6. 6.

    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)

    Google Scholar 

  7. 7.

    Higueras, I., März, R., Tischendorf, C.: Stability preserving integration of index-1 DAEs. Appl. Numer. Math. 45(2–3), 175–200 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Kunkel, P., Mehrmann, V.: Differential-Algebraic Aquations. Analysis and Numerical Solution. European Mathemantical Society, Zürich (2006)

    Google Scholar 

  9. 9.

    Kunkel, P., Mehrmann, V.: Stability properties of differential-algebraic equations and spin-stabilized discretization. Electron. Trans. Numer. Anal. 26, 385–420 (2007)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    März, R.: On one-leg methods for differential-algebraic equations. Circuits Systems Signal Process 5(1), 87–95 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Springer, Heidelberg (2013)

    Google Scholar 

  12. 12.

    Linh, V.H., Mehrmann, V.: Approximation of spectral intervals and associated leading directions for linear differential-algebraic equations via smooth singular value decompositions. SIAM J. Numer. Anal. 49(5), 1810–1835 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Linh, V.H., Mehrmann, V.: Efficient integration of matrix-valued non-stiff DAEs by half-explicit methods. J. Comput. Appl. Math. 262, 346–360 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Linh, V.H., Mehrmann, V., Van Vleck, E.: QR Methods and error analysis for computing Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations. Adv. Comput. Math. 35(2–4), 281–322 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Linh, V.H., Truong, N.D.: Runge-kutta methods revisited for a class of structured strangeness-free DAEs. Electron. Trans. Numer. Analysis 48, 131–155 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Liniger, W.: Multistep and one-leg methods for implicit mixed differential algebraic systems. IEEE Trans. Circ. and Syst. 26, 755–762 (1979)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referee for the very helpful comments and suggestions that led to the improvements of this paper.

Funding

This work was supported by the Nafosted Project No. 101.02-2017.314.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vu Hoang Linh.

Additional information

Dedicated to Professor Pham Ky Anh on the occasion of his 70th birthday

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Linh, V.H., Truong, N.D. Stable Numerical Solution for a Class of Structured Differential-Algebraic Equations by Linear Multistep Methods. Acta Math Vietnam 44, 955–976 (2019). https://doi.org/10.1007/s40306-018-00310-5

Download citation

Keywords

  • Differential-algebraic equations
  • Strangeness-free form
  • Linear multistep methods
  • Convergence
  • Stability

Mathematics Subject Classification (2010)

  • 65L80
  • 65L05
  • 65L06
  • 65L20