Skip to main content
SpringerLink
Log in

Optimal first- to sixth-order accurate Runge-Kutta schemes

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

An optimal choice of free parameters in explicit Runge-Kutta schemes up to the sixth order is discussed. A sixth-order seven-stage scheme that is immediately ahead of Butcher’s second barrier is constructed. The study is performed in the most general form, and its results are applicable to both autonomous and nonautonomous 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. E. Hairer, S. P. Nörsett, and G. Wanner, Solving Ordinary Differential Equations. I: Nonstiff Problems (Springer-Verlag, Berlin, 1987; Mir, Moscow, 1990).

    MATH  Google Scholar 

  2. J. C. Butchern, “Coefficients for Study of Runge-Kutta Integration Processes,” J. Austral. Math. Soc. 3, 185–201 (1963).

    Article  MathSciNet  Google Scholar 

  3. G. M. Hammud, “Three-Dimensional Family of Seven-Stage Runge-Kutta Methods of the Sixth Order,” Vychisl. Metody Program. 2(2), 71–78 (2001).

    Google Scholar 

  4. K. Dekker and J. G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations (North-Holland, Amsterdam, 1984; Mir, Moscow, 1988).

    MATH  Google Scholar 

  5. G. I. Marchuk and V. V. Shaidurov, Improving the Accuracy of Finite Difference Schemes (Nauka, Moscow, 1979) [in Russian].

    Google Scholar 

  6. N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, and B. V. Rogov, Computations with Quasi-Equidistant Grids (Fizmatlit, Moscow, 2005) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematical Modeling, Russian Academy of Sciences, Miusskaya pl. 4a, Moscow, 125047, Russia

    E. A. Alshina & N. N. Kalitkin

  2. Moscow State Institute of Electronic Engineering (Technical University), Zelenograd, Moscow, 124498, Russia

    E. M. Zaks

Authors
  1. E. A. Alshina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. M. Zaks
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. N. Kalitkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. A. Alshina.

Additional information

Original Russian Text © E.A. Alshina, E.M. Zaks, N.N. Kalitkin, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 3, pp. 418–429.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alshina, E.A., Zaks, E.M. & Kalitkin, N.N. Optimal first- to sixth-order accurate Runge-Kutta schemes. Comput. Math. and Math. Phys. 48, 395–405 (2008). https://doi.org/10.1134/S0965542508030068

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542508030068

Keywords

Search

Navigation