Skip to main content
SpringerLink
Log in

Towards explicit methods for differential algebraic equations

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

Explicit methods have previously been proposed for parabolic PDEs and for stiff ODEs with widely separated time constants. We discuss ways in which Differential Algebraic Equations (DAEs) might be regularized so that they can be efficiently integrated by explicit methods. The effectiveness of this approach is illustrated for some simple index three problems.

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. A. Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput., 25 (2002), pp. 2041–2054.

    Article  MathSciNet  Google Scholar 

  2. C. Arévalo, C. Führer, and G. Söderlind, Regular and singular β-blocking of difference corrected multistep methods for nonstiff index-2 DAEs, Appl. Numer. Math., 35 (2000), pp. 293–305.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Arnold, Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2, BIT, 38 (1998), pp. 415–438.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Arnold and A. Murua, Non-stiff integrators for differential-algebraic systems of index 2, Numer. Algorithms, 19 (1998), pp. 25–41.

    Article  MathSciNet  MATH  Google Scholar 

  5. U. M. Ascher, H. Chin, and S. Reich, Stabilization of DAEs and invariant manifolds, Numer. Math., 67 (1994), pp. 131–149.

    Article  MathSciNet  MATH  Google Scholar 

  6. U. M. Ascher and L. R. Petzold, Projected implicit Runge–Kutta methods for differential-algebraic equations, SIAM J. Numer. Anal., 28 (1991), pp. 1097–1120.

    Article  MathSciNet  MATH  Google Scholar 

  7. T. R. Bashkow, The A matrix, new network description, IRE Trans. Circuit Theory, CT-4 (1957), pp. 117–120.

    Google Scholar 

  8. J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comp. Math. Appl. Mech. Eng., 1 (1976), pp. 1–16.

    Article  MathSciNet  Google Scholar 

  9. Y. Cao and Q. Li, Highest order multistep formula for solving index-2 differential algebraic equations, BIT, 38 (1998), pp. 663–673.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Eriksson, C. Johnson, and A. Logg, Explicit time-stepping for stiff ODEs, SIAM J. Sci. Comput., 25 (2003), pp. 1142–1157.

    Article  MathSciNet  MATH  Google Scholar 

  11. C. W. Gear, Simultaneous numerical solution of differential-algebraic equations, IEEE Trans. Circuit Theory, CT-18 (1971), pp. 89–95.

    Article  Google Scholar 

  12. C. W. Gear and I. G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comput., 24 (2003), pp. 1091–1106.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. W. Gear and I. G. Kevrekidis, Telescopic projective methods for parabolic differential equations, J. Comput. Phys., 187 (2003), pp. 95–109.

    Article  MathSciNet  MATH  Google Scholar 

  14. G. D. Hachtel, R. K. Brayton, and F. G. Gustavson, The sparse tableau approach to network analysis and design, IEEE Trans. Circuit Theory, CT-18 (1971), pp. 101–113.

    Article  Google Scholar 

  15. E. Hairer, Symmetric projection methods for differential equations on manifolds, BIT, 40 (2000), pp. 726–735.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Hanke, A note on the consistent initialization for non-lineaar index-2 differential-algebraic equations in Hessenberg form, Report 2004:02, Parallel and Scientific Computing Institute, Roy. Inst. Tech., Stockholm, Sweden, 2004.

  17. E. S. Kuh and R. A. Rohrer, The state-variable approach to network analyis, Proc. IEEE, 53 (1965), pp. 672–686.

    Article  Google Scholar 

  18. C. Lubich, U. Nowak, U. Pohle, and Ch. Engstler, MEXX – Numerical software for the integration of constrained mechanical multibody systems, Tech. report SC92-12, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, 1992.

  19. R. Marz and C. Tischendorf, Recent results in solving index-2 differential-algebraic equations in circuit simulation, SIAM J. Sci. Comput., 18 (1997), pp. 139–159.

    Article  MathSciNet  Google Scholar 

  20. A. A. Medovikov, High order explicit methods for parabolic equations, BIT, 38 (1998), pp. 372–390.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. P. Sommeijer, L. F. Shampine, and J. G. Verwer, RKC: An explicit solver for parabolic PDEs, J. Comput. Appl. Math., 88 (1998), pp. 315–326.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. J. van der Houwen, The development of Runge–Kutta methods for partial differential equations, Appl. Num. Math., 20 (1996), pp. 261–272.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 17 Honey Brook Drive, Princeton, NJ, 08540, USA

    C. W. Gear

Authors
  1. C. W. Gear
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. W. Gear.

Additional information

In memory of Germund Dahlquist (1925–2005).

AMS subject classification (2000)

65-L80, 34-04

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gear, C. Towards explicit methods for differential algebraic equations . Bit Numer Math 46, 505–514 (2006). https://doi.org/10.1007/s10543-006-0068-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-006-0068-x

Key words

Search

Navigation