Skip to main content
SpringerLink
Log in

Hybrid BDF methods for the numerical solutions of ordinary differential equations

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

Abstract

In this article, we have presented the details of hybrid methods which are based on backward differentiation formula (BDF) for the numerical solutions of ordinary differential equations (ODEs). In these hybrid BDF, one additional stage point (or off-step point) has been used in the first derivative of the solution to improve the absolute stability regions. Stability domains of our presented methods have been obtained showing that all these new methods, we say HBDF, of order p, p = 2,4,..., 12, are A(α)-stable whereas they have wide stability regions comparing with those of some known methods like BDF, extended BDF (EBDF), modified EBDF (MEBDF), adaptive EBDF (A-EBDF), and second derivtive Enright methods. Numerical results are also given for five test 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. Yousuf, M.: On the class of high order time stepping schemes based on Pade approximations for the numerical solution of Burgers’ equation. Appl. Math. Comput. 205, 442–453 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Psihoyios, G.: Solving time dependent PDEs via an improved modified extended BDF scheme. Appl. Math. Comput. 184, 104–115 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Hojjati, G., et al.: New second derivative multistep methods for stiff systems. Appl. Math. Model. 30, 466–476 (2006)

    Article  MATH  Google Scholar 

  4. Butcher, J.C., Wright, W.M.: Appplications of doubly companian matrices. Appl. Numer. Math. 56, 358–373 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Huang, S.J.Y.: Implementation of general linear methods for stiff ordinery differential equations. Ph.D thesis, Department of Mathematics, Auckland University (2005)

  6. Hojjati, G., et al.: A- EBDF: an adaptivemethod for numerical solution of stiff systems of ODEs. Math. Comput. Simul. 66, 33–41 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kelley, C.T.: Solving Nonlinear Equations with Newton’s Method, Fundamentals of Algorithms. SIAM, Philadelphia (2003)

    Google Scholar 

  8. Cash, J.R.: Modified extended backward differentiation formulae for the numerical solution of stiff inintial value problems in ODEs and DAEs. J. CAM 125, 117–130 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Li, S.: Stability and B-convergence properties of multistep Runge-Kutta methods. Math. Comput. 69(232), 1481–1504 (1999)

    Article  Google Scholar 

  10. Wu, X.-Y.: A sixth-order A-stable explicit one-step method for stiff systems. Comput. Math. Appl. 35(9), 59–64 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Butcher, J.C., Diamantakis, M.: DESIRE: diagonally extended singly-implicit Runge-Kutta effective order methods. Numer. Algorithms 17, 121–145 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Butcher, J.C., Cash, J.R., Diamantakis, M.: DESI methods for stiff initial value problems. ACM Trans. Math. Softw. 22, 401–422 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problem. Springer, Berlin (1996)

    Google Scholar 

  14. Cash, J.R.: The MOL solution of time dependant partial differential equations. Comput. Math. Appl. 31(11), 69–78 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Diamantakis, M.: Digonally extended singly-implicit Runge-Kutta methods for stiff IVPs. Ph.D. thesis, Imperial College, London (1995)

  16. Cash, J.R., Considine, S.: An MEBDF code for stiff initial value problem. ACM Trans. Math. Softw. 18, 142–160 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Springer, New York (1992)

    Google Scholar 

  18. Cash, J.R.: On the integration of stiff systems of ODEs using modified extended backward differentiation formula. Comput. Math. Appl. 9, 645–657 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. England, R.: Some hybrid implicit stiffly stable methods for ordinary differential equations. In: Numerical Analysis, Cocoyoc 1981. Lecture Notes in Maths, vol. 909, pp. 147–158. Springer, New York (1982)

    Chapter  Google Scholar 

  20. Cash, J.R.: On the integration of stiff systems of ODEs using extended backward differentiation formula. Numer. Math. 34(2), 235–246 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Alexander. R.: Diagonally implicit Runge-Kutta methods for stiff ODEs. SIAM J. Numer. Anal. 14, 1006–10021 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lambert, J.D.: Computational Methods in Ordinary Differential Equations, pp. 143–144. Wiley, London (1972)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pune, Pune, India

    Moosa Ebadi

  2. Department of Mathematics, M.I.T. College, Pune, India

    M. Y. Gokhale

Authors
  1. Moosa Ebadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Y. Gokhale
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moosa Ebadi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ebadi, M., Gokhale, M.Y. Hybrid BDF methods for the numerical solutions of ordinary differential equations. Numer Algor 55, 1–17 (2010). https://doi.org/10.1007/s11075-009-9354-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-009-9354-4

Keywords

Search

Navigation