Skip to main content
SpringerLink
Log in

A local-global version of a stepsize control for Runge-Kutta methods

  • Published:
Korean Journal of Computational & Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper we develop a new procedure to control stepsize for Runge-Kutta methods applied to both ordinary differential equations and semi-explicit index 1 differential-algebraic equations. In contrast to the standard approach, the error control mechanism presented here is based on monitoring and controlling both the local and global errors of Runge-Kutta formulas. As a result, Runge-Kutta methods with the local-global stepsize control solve differential or differential-algebraic equations with any prescribed accuracy (up to round-off errors).

For implicit Runge-Kutta formulas we give the sufficient number of both full and modified Newton iterations allowing the iterative approximations to be correctly used in the procedure of the local-global stepsize control. In addition, we develop a stable local-global error control mechanism which is applicable for stiff problems. Numerical tests support the theoretical results of the paper.

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. C. Arévalo, G. Söderlind,Convergence of multistep discretizations of DAEs, BIT35 (1995), 143–168.

    Article  MATH  MathSciNet  Google Scholar 

  2. E. Eich, C. Führer, J. Yen,On the error control for multistep methods applied to ODEs with invariants and DAEs in multibody dynamics, Mech. Struct. and Mach.23(2) (1995), 159–179.

    Article  Google Scholar 

  3. C.W. Gear, L.R. Petzold,ODE methods for the solution of differential/algebraic systems, SIAM J. Numer. Anal.21 (1984), pp. 716–728.

    Article  MATH  MathSciNet  Google Scholar 

  4. E. Hairer, S.P. Nørsett, G. Wanner,Solving ordinary differential equations I: Nonstiff problems, Springer-Verlag, Berlin, 1987, 1993.

    MATH  Google Scholar 

  5. E. Hairer, Ch. Lubich, M. Roche,The numerical solution of differential-algebraic systems by Runge-Kutta methods, Lecture Note in Math. 1409, Springer-Verlag, Berlin, 1989.

    MATH  Google Scholar 

  6. E. Hairer, G. Wanner,Solving ordinary differential equations II: Stiff and differentialalgebraic problems, Springer-Verlag, Berlin, 1991, 1996.

    Google Scholar 

  7. K.R. Jackson, A. Kv→rnø, S.P. Nørsett,The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas, Applied Numerical Mathematics15 (1994), 341–356.

    Article  MATH  MathSciNet  Google Scholar 

  8. G.Yu. Kulikov,A method for the numerical solution of the autonomous Cauchy problem with an algebraic restriction of the phase variables, (in Russian) Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., (1992), No. 1, 14–19;translation in Moscow Univ. Math. Bull.47 (1992), No. 1, 14–18.

  9. G.Yu. Kulikov,The numerical solution of an autonomous Cauchy problem with an algebraic constraint on the phase variables (the nonsingular case), (in Russian) Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., (1993), No. 3, 6–10;translation in Moscow Univ. Math. Bull.48 (1993), No. 3, 8–12.

  10. G.Yu. Kulikov,The numerical solution of the autonomous Cauchy problem with an algebraic relation between the phase variables, (in Russian) Zh. Vychisl. Mat. Mat. Fiz.,33 (1993), No. 4, 522–540;translation in Comp. Maths Math. Phys.,33 (1993), No. 4, 477–492.

    MATH  MathSciNet  Google Scholar 

  11. G.Yu. Kulikov and P.G. Thomsen,Convergence and implementation of implicit Runge-Kutta methods for DAEs, Technical report 7/1996, IMM, Technical University of Denmark, Lyngby, 1996.

    Google Scholar 

  12. G.Yu. Kulikov,Convergence theorems for iterative Runge-Kutta methods with a constant integration step, (in Russian) Zh. Vychisl. Mat. Mat. Fiz.,36 (1996), No. 8, 73–89;translation in Comp. Maths Math. Phys.,36 (1996), No. 8, 1041–1054.

    MathSciNet  Google Scholar 

  13. G.Yu. Kulikov,Numerical methods solving the semi-explicit differential-algebraic equations by implicit multistep fixed stepsize methods, Korean J. Comput. & Appl. Math.4 (1997), No. 2, 281–318.

    MATH  MathSciNet  Google Scholar 

  14. G.Yu. Kulikov,On the numerical solution of the Cauchy problem for a system of differential-algebraic equations by means of implicit Runge-Kutta methods with a variable integration step, (in Russian) Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., (1997), No. 5, 7–11;translation in Moscow Univ. Math. Bull.,52 (1997), No. 5, 6–10.

  15. G.Yu. Kulikov,A theory of symmetric one-step methods for differential-algebraic equations, Russian J. Numer. Anal. Math. Modelling,12 (1997), No. 6, 501–523.

    Article  MATH  MathSciNet  Google Scholar 

  16. G.Yu. Kulikov,Numerical solution of the Cauchy problem for a system of differentialalgebraic equations with the use of implicit Runge-Kutta methods with a nontrivial predictor, (in Russian) Zh. Vychisl. Mat. Mat. Fiz.,38 (1998), No. 1, 68–84;translation in Comp. Maths Math. Phys.,38 (1998), No. 1, 64–80.

    MathSciNet  Google Scholar 

  17. G.Yu. Kulikov,A local-global stepsize control for multistep methods applied to semiexplicit index 1 differential-algebraic equations, Korean J. Comput. & Appl. Math.6 (1999), No. 3, 463–492.

    MATH  MathSciNet  Google Scholar 

  18. A. Kværnø,The order of Runge-Kutta methods applied to semi-explicit DAEs of index 1, using Newton-type iterations to compute the internal stage values, Technical report 2/1992, Mathematical Sciences Div., Norwegian Institute of Technology, Trondheim, 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Cybernetics, Faculty of Mechanics and Mathematics, Ulyanovsk State University, L.Tolstoy Str. 42, 432700, Ulyanovsk, Russia

    G. Y. Kulikov

Authors
  1. G. Y. Kulikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Y. Kulikov.

Additional information

This work was supported in part by the Russian Academy of Sciences, the Ministry of General and Professional Education of Russia (Scientific Program ”Universities of Russia — Basic Research”, project No. 230) and the Russian Foundation of the Basic Research (project No. 98-01-00006).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kulikov, G.Y. A local-global version of a stepsize control for Runge-Kutta methods. Korean J. Comput. & Appl. Math. 7, 289–318 (2000). https://doi.org/10.1007/BF03012194

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03012194

AMS Mathematics Subject Classifications

Key word and phrases

Search

Navigation