Skip to main content
SpringerLink
Log in

Runge-Kutta schemes and numerical instabilities: The logistic equation

  • Conference paper
  • First Online:
Differential Equations and Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1285))

  • 809 Accesses

Abstract

We consider the Logistic differential equation and several finite difference schemes that can be used to numerically integrate the equation. We show that linear stability analysis about the fixed points of the finite difference schemes allow an understanding of how and when instabilities can occur. Some rules for modeling differential equations by finite difference schemes are presented.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Yamaguti and S. Ushiki, "Chaos in Numerical Analysis of Ordinary Differential Equations," Physica 3D (1981), 618–626.

    MathSciNet  MATH  Google Scholar 

  2. S. Ushiki, "Central Difference Scheme and Chaos," Physica 4D (1982), 407–424.

    MathSciNet  MATH  Google Scholar 

  3. R. E. Mickens, "Difference Equations Models of Differential Equations Having Zero Local Truncation Error," pps. 445–449, in Differential Equations, edited by I. W. Knowles and R. T. Lewis (North-Holland, Amsterdam, 1984).

    Google Scholar 

  4. A. R. Mitchell and J. C. Bruch, Jr., "A Numerical Study of Chaos in a Reaction-Diffusion Equation," Numerical Methods for Partial Differential Equations 1 (1985), 13–23.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. M. Sanz-Serna, "Studies in Numerical Instability I. Why Do Leapfrog Schemes Go Unstable?," SIAM Journal of Scientific and Statistical Computing 6 (1985), 923–938.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Atlanta University, 30314, Atlanta, Georgia

    Ronald E. Mickens

Authors
  1. Ronald E. Mickens
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ian W. Knowles Yoshimi Saitō

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Mickens, R.E. (1987). Runge-Kutta schemes and numerical instabilities: The logistic equation. In: Knowles, I.W., Saitō, Y. (eds) Differential Equations and Mathematical Physics. Lecture Notes in Mathematics, vol 1285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0080612

Download citation

  • DOI: https://doi.org/10.1007/BFb0080612

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18479-9

  • Online ISBN: 978-3-540-47983-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation