Skip to main content
SpringerLink
Log in

Second derivative general linear methods with inherent Runge–Kutta stability

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

Abstract

In this paper, we find some relationships among the coefficients matrices of second derivative general linear methods (SGLMs) which are sufficient conditions, but not necessary, to ensure the methods have Runge–Kutta stability (RKS) property. Considering these conditions, we construct some A– and L–stable SGLMs with inherent RKS of orders up to five. Also, some numerical experiments for the constructed methods in variable stepsize environment are given.

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. Abdi, A., Bras, M., Hojjati, G.: On the construction of second derivative diagonally implicit multistage integration methods. Appl. Numer. Math. 76, 1–18 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abdi, A., Hojjati, G.: An extension of general linear methods. Numer. Algor. 57, 149–167 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abdi, A., Hojjati, G.: Maximal order of second derivative general linear methods with Runge–kutta stability. Appl. Numer. Math. 61, 1046–1058 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abdi, A., Hojjati, G.: Implementation of Nordsieck second derivative methods for stiff ODEs. Appl. Numer. Math. 94, 241–253 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Butcher, J.C.: On the convergence of numerical solutions to ordinary differential equations. Math. Comp. 20, 1–10 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  6. Butcher, J.C.: General linear methods for stiff differential equations. BIT 41, 240–264 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Butcher, J.C.: Numerical methods for ordinary differential equations. Wiley, New York (2008)

    Book  MATH  Google Scholar 

  8. Butcher, J.C., Hojjati, G.: Second derivative methods with RK stability. Numer. Algor. 40, 415–429 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Butcher, J.C., Wright, W.M.: A transformation relating explicit and diagonally-implicit general linear methods. Appl. Numer. Math. 44, 313–327 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Cash, J.R.: Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J. Numer. Anal. 18, 21–36 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chakravarti, P.C., Kamel, M.S.: Stiffly stable second derivative multistep methods with higher order and improved stability regions. BIT 23, 75–83 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chan, R.P.K., Tsai, A.Y.J.: On explicit two-derivative Runge–Kutta methods. Numer. Algor. 53, 171–194 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  16. Enright, W.H.: Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–331 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations I: nonstiff problems. Springer, Berlin (2000)

    MATH  Google Scholar 

  18. Hojjati, G., Rahimi Ardabili, M.Y., Hosseini, S.M.: New second derivative multistep methods for stiff systems. Appl. Math. Model. 30, 466–476 (2006)

    Article  MATH  Google Scholar 

  19. Huang, S.J.Y.: Implementation of general linear methods for stiff ordinary differential equations. PhD thesis, Department of Mathematics Auckland University (2005)

  20. Jackiewicz, Z.: General linear methods for ordinary differential equations. Wiley, New Jersey (2009)

    Book  MATH  Google Scholar 

  21. Lee, J.H.J.: Numerical methods for ordinary differential equations: a survey of some standard methods. MSc thesis, Department of Mathematics Auckland University (2004)

  22. Schäfer, E.: A new approach to explain the High Irradiance Responses of photomorphogenesis on the basis of phytochrome. J. Math. Biology. 2, 41–56 (1975)

    Article  MATH  Google Scholar 

  23. Wright, W.M.: Explicit general linear methods with inherent Runge–Kutta Stability. Numer. Algor. 31, 381–399 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wright, W.M.: General linear methods with inherent Runge–Kutta stability. Phd thesis, University of Auckland (2003)

Download references

Author information

Authors and Affiliations

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

    A. Movahedinejad, G. Hojjati & A. Abdi

Authors
  1. A. Movahedinejad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Hojjati
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Abdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Hojjati.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Movahedinejad, A., Hojjati, G. & Abdi, A. Second derivative general linear methods with inherent Runge–Kutta stability. Numer Algor 73, 371–389 (2016). https://doi.org/10.1007/s11075-016-0099-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0099-6

Keywords

Search

Navigation