Skip to main content
SpringerLink
Log in

Two-step Runge-Kutta Methods with Quadratic Stability Functions

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We describe the construction of implicit two-step Runge-Kutta methods with stability properties determined by quadratic stability functions. We will aim for methods which are A-stable and L-stable and such that the coefficients matrix has a one point spectrum. Examples of methods of order up to eight are provided.

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. Bartoszewski, Z., Jackiewicz, Z.: Construction of two-step Runge-Kutta methods of high order for ordinary differential equations. Numer. Algorithms 18, 51–70 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bartoszewski, Z., Jackiewicz, Z.: Nordsieck representation of two-step Runge-Kutta methods for ordinary differential equations. Appl. Numer. Math. 53, 149–163 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bartoszewski, Z., Jackiewicz, Z.: Derivation of continuous explicit two-step Runge-Kutta methods of order three. J. Comput. Appl. Math. 205, 764–776 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bartoszewski, Z., Podhaisky, H., Weiner, R.: Construction of stiffly accurate two-step Runge-Kutta methods of order three and their continuous extensions. Report 07-01, FB Mathematik und Informatik, Martin-Luther-Universität, Halle-Wittenberg (2007)

  5. Bellen, A., Jackiewicz, Z., Zennaro, M.: Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods. BIT 32, 104–117 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burrage, K.: A special family of Runge-Kutta methods for solving stiff differential equations. BIT 18, 22–41 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  7. Burrage, K., Butcher, J.C., Chipman, F.H.: An implementation of singly implicit Runge-Kutta methods. BIT 20, 326–340 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  8. Butcher, J.C.: A transformed implicit Runge-Kutta method. J. Assoc. Comput. Mach. 26, 731–738 (1979)

    MATH  MathSciNet  Google Scholar 

  9. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. Wiley, New York (1987)

    MATH  Google Scholar 

  10. Butcher, J.C.: Diagonally-implicit multi-stage integration methods. Appl. Numer. Math. 11, 347–363 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (2003)

    Book  MATH  Google Scholar 

  12. Butcher, J.C., Jackiewicz, Z.: Construction of high order diagonally implicit multistage integration methods for ordinary differential equations. Appl. Numer. Math. 27, 1–12 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Butcher, J.C., Tracogna, S.: Order conditions for two-step Runge-Kutta methods. Appl. Numer. Math. 24, 351–364 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Butcher, J.C., Wright, W.M.: Applications of doubly companion matrices. Appl. Numer. Math. 56, 358–373 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chollom, J., Jackiewicz, Z.: Construction of two-step Runge-Kutta methods with large regions of absolute stability. J. Comput. Appl. Math. 157, 125–137 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. D’Ambrosio, R.: Highly stable multistage numerical methods for Functional Equations: theory and implementation issues. Ph.D. thesis, University of Salerno (2010)

  19. D’Ambrosio, R., Izzo, G., Jackiewicz, Z.: Search for highly stable two-step Runge–Kutta methods for ordinary differential equations. Appl. Numer. Math. (accepted)

  20. D’Ambrosio, R., Jackiewicz, Z.: Construction and implementation of highly stable two-step collocation methods (submitted)

  21. Hairer, E., Wanner, G.: Order conditions for general two-step Runge-Kutta methods. SIAM J. Numer. Anal. 34, 2087–2089 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (1997)

    Google Scholar 

  23. Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, New York (2009)

    Book  MATH  Google Scholar 

  24. Jackiewicz, Z., Tracogna, S.: A general class of two-step Runge-Kutta methods for ordinary differential equations. SIAM J. Numer. Anal. 32, 1390–1427 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Jackiewicz, Z., Tracogna, S.: Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations. Numer. Algorithms 12, 347–368 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Jackiewicz, Z., Verner, J.H.: Derivation and implementation of two-step Runge-Kutta pairs. Japan J. Ind. Appl. Math. 19, 227–248 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, New York (1973)

    MATH  Google Scholar 

  28. Schur, J.: Über Potenzreihen die im Innern des Einheitskreises beschränkt sind. J. Reine Angew. Math. 147, 205–232 (1916)

    Google Scholar 

  29. Tracogna, S.: Implementation of two-step Runge-Kutta methods for ordinary differential equations. J. Comput. Appl. Math. 76, 113–136 (1997)

    Article  MathSciNet  Google Scholar 

  30. Tracogna, S., Welfert, B.: Two-step Runge-Kutta: Theory and practice. BIT 40, 775–799 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

  32. Wright, W.M.: Explicit general linear methods with inherent Runge-Kutta stability. Numer. Algorithms 31, 381–399 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Salerno, Fisciano (Sa), Italy

    D. Conte & R. D’Ambrosio

  2. Department of Mathematics, Arizona State University, Tempe, AZ, USA

    Z. Jackiewicz

  3. AGH University of Science and Technology, Kraków, Poland

    Z. Jackiewicz

Authors
  1. D. Conte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. D’Ambrosio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Jackiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. D’Ambrosio.

Additional information

The work of R. D’Ambrosio was partially supported by the University of Salerno, according to an exchange agreement for bi-nationally supervised Ph.D. thesis.

The work of Z. Jackiewicz was partially supported by the National Science Foundation under grant NSF DMS–0510813.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Conte, D., D’Ambrosio, R. & Jackiewicz, Z. Two-step Runge-Kutta Methods with Quadratic Stability Functions. J Sci Comput 44, 191–218 (2010). https://doi.org/10.1007/s10915-010-9378-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-010-9378-x

Keywords

Search

Navigation