Skip to main content
SpringerLink
Log in

Global error estimation for explicit general linear methods

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We describe an approach to global error estimation for explicit general linear methods. This approach is based on computation of two numerical solutions by pairs of general linear methods of the same order and stage order and with proportional global error functions. The results of numerical experiments indicate that this approach is very accurate in constant and variable stepsize environments.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Butcher, J.C., Jackiewicz, Z.: A new approach to error estimation for general linear methods. Numer. Math. 95, 487–502 (2003)

  2. Butcher, J.C., Jackiewicz, Z.: Construction of general linear methods with Runge–Kutta stability properties. Numer. Algorithms 36, 53–72 (2004)

  3. Butcher, J.C., Wright, W.M.: The construction of practical general linear methods. BIT 43, 695–721 (2003)

  4. Constantinescu, E.M.: Generalizing global error estimation for ordinary differential equations by using coupled time-stepping methods. J. Comput. Appl. Math. 332, 140–158 (2018)

  5. Hairer, E., Lubich, C.: Asymptotic expansions of the global error of fixed-stepsize methods. Numer. Math. 45, 345–360 (1984)

  6. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, Berlin, Heidelberg, New York (1993)

  7. Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. John Wiley, Hoboken, New Jersey (2009)

  8. Kulikov, G.: Cheap global error estimation in some Runge-Kutta pairs. IMA J. Numer. Anal. 33, 136–163 (2013)

  9. Skeel, R.: Analysis of fixed-stepsize methods. SIAM J. Numer. Anal. 13, 664–685 (1976)

  10. Wright, W.: General linear methods with inherent Runge–Kutta stability. Ph.D. thesis, The University of Auckland, New Zealand (2002)

Download references

Acknowledgements

The authors wish to express their gratitude to the anonymous reviewers for their useful comments which helped improve the presentation of this paper.

Author information

Authors and Affiliations

  1. Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

    Ali Abdi & Gholamreza Hojjati

  2. Dipartimento di Matematica e Applicazioni “R.Caccioppoli”, Università di Napoli Federico II, Napoli, Italy

    Giuseppe Izzo

  3. School of Mathematical & Statistical Sciences, Arizona State University, Tempe, Arizona, 85287, USA

    Zdzislaw Jackiewicz

  4. Department of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland

    Zdzislaw Jackiewicz

Authors
  1. Ali Abdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gholamreza Hojjati
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giuseppe Izzo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zdzislaw Jackiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali Abdi.

Additional information

Publisher’s note

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

The work of the first and second authors was supported by the University of Tabriz, International and Academic Cooperation Directorate, in the framework of TabrizU-300 program.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abdi, A., Hojjati, G., Izzo, G. et al. Global error estimation for explicit general linear methods. Numer Algor 89, 1075–1093 (2022). https://doi.org/10.1007/s11075-021-01146-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01146-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation