Advertisement

Numerical Algorithms

, Volume 43, Issue 1, pp 99–121 | Cite as

One-leg variable-coefficient formulas for ordinary differential equations and local–global step size control

  • Gennady Yu. Kulikov
  • Sergey K. Shindin
Article

Abstract

In this paper we discuss a class of numerical algorithms termed one-leg methods. This concept was introduced by Dahlquist in 1975 with the purpose of studying nonlinear stability properties of multistep methods for ordinary differential equations. Later, it was found out that these methods are themselves suitable for numerical integration because of good stability. Here, we investigate one-leg formulas on nonuniform grids. We prove that there exist zero-stable one-leg variable-coefficient methods at least up to order 11 and give examples of two-step methods of orders 2 and 3. In this paper we also develop local and global error estimation techniques for one-leg methods and implement them with the local–global step size selection suggested by Kulikov and Shindin in 1999. The goal of this error control is to obtain automatically numerical solutions for any reasonable accuracy set by the user. We show that the error control is more complicated in one-leg methods, especially when applied to stiff problems. Thus, we adapt our local–global step size selection strategy to one-leg methods.

Keywords

one-leg methods local error estimation global error estimation stiff problems 

Mathematics Subject Classifications

65L06 65L20 65L70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2003)zbMATHCrossRefGoogle Scholar
  2. 2.
    Crouzeix, M., Lisbona, F.J.: The convergence of variable-stepsize, variable formula, multistep methods. SIAM J. Numer. Anal. 21, 512–534 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Dahlquist, G.: Error analysis of a class of methods for stiff non-linear initial value problems. Lecture Notes in Mathematics vol. 506, pp. 60–74. Springer, Berlin Heidelberg New York (1975)Google Scholar
  5. 5.
    Dahlquist, G.: G-stability is equivalent to A-stability. BIT 18, 384–401 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Dahlquist, G.: On the local and global errors of one-leg methods. Report TRITA-NA-8110 Royal Institute of Technology, Stockholm (1981)Google Scholar
  7. 7.
    Dahlquist, G.: On one-leg multistep methods. SIAM J. Numer. Anal. 20, 1130–1138 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dahlquist, G., Liniger, W., Nevanlinna, O.: Stability of two-step methods for variable integration steps. SIAM J. Numer. Anal. 20, 1071–1085 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dekker, K., Verwer, J.G.: Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations. North-Holland, Amsterdam, The Netherlands (1984)Google Scholar
  10. 10.
    Grigorieff, R.D.: Stability of multistep methods on variable grids. Numer. Math. 42, 359–377 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin Heidelberg New York (1993)Google Scholar
  12. 12.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. Springer, Berlin Heidelberg New York (1996)zbMATHGoogle Scholar
  13. 13.
    Hundsdorfer, W.H., Steininger, B.I.: Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems. BIT 31, 124–143 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kulikov, G.Yu., Shindin, S.K.: A technique for controlling the global error in multistep methods. Zh. Vychisl. Mat. Mat. Fiz. (in Russian) 40, 1308–1329 (2000); translation in Comput. Math. Math. Phys. 40, 1255–1275 (2000)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Kulikov, G.Yu., Shindin, S.K.: On effective computation of asymptotically correct estimates of the local and global errors for multistep methods with fixed coefficients. Zh. Vychisl. Mat. Mat. Fiz. (in Russian) 44, 847–868 (2004); translation in Comput. Math. Math. Phys. 44, 794–814 (2004)Google Scholar
  16. 16.
    Kulikov, G.Yu., Shindin, S.K.: On interpolation-type multistep methods with automatic global error control. Zh. Vychisl. Mat. Mat. Fiz. (in Russian) 44, 1400–1421 (2004); translation in Comput. Math. Math. Phys. 44, 1314–1333 (2004)Google Scholar
  17. 17.
    Kulikov, G.Yu., Shindin, S.K.: One-leg integration of ordinary differential equations with global error control. Comput. Methods Appl. Math. 5, 86–96 (2005)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Kulikov, G.Yu., Shindin, S.K.: Global error estimation and extrapolated multistep methods for index 1 differential-algebraic systems. BIT 45, 517–542 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Nevanlinna, O., Liniger, W.: Contractive methods for stiff differential equations. Part I. BIT 18, 457–474 (1978); Part II BIT 19, 53–72 (1979)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.School of Computational and Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

Personalised recommendations