Linear k-step method (k2) with constant coefficients are derived in a “natural” way by choosing as the second characteristic polynomial a Schur polynomial whose coefficients depend on a certain set of parameters. The choice of these parameters is based on a result by Marden concerning the location of the zeros of a class of rational functions. For the (practically important) case k = 2 it is shown that the resulting class of methods is A -stable and has order p = 2. The trapezoidal rule and a class of one-step methods introduced by Lininger and Willoughby turn out to be degenerate cases of this class of two-step-methods.


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  1. 1.
    Bjurel, G. et al; Survey of stiff ordinary differential equations.Report NA 70.11, Royal Institute of Technology, Stockholm, 1970.Google Scholar
  2. 2.
    Curtiss, C.F. and J.O. Hirschfelder: Integration of stiff equations. Proc.Nat. Acad.Sci. USA, 38 (1952). 235–243.CrossRefGoogle Scholar
  3. 3.
    Dahlquist, G.: A special stability problem for linear multistep methods. BIT, 3 (1963), 27–43.CrossRefGoogle Scholar
  4. 4.
    Duffin, R.J.: Algorithms for classical stability problems. SIAM Review, 11 (1969), 196–213.CrossRefGoogle Scholar
  5. 5.
    Gourlay, A.R.: A note on trapezoidal methods for the solution of initial value problems. Math.Comp. 24 (1970), 629–633.CrossRefGoogle Scholar
  6. 6.
    Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York, 1962.Google Scholar
  7. 7.
    Lindberg, B.: On Smoothing and extrapolation for the trapezoidal rule. BIT 11 (1971), 29–52.CrossRefGoogle Scholar
  8. 8.
    Liniger, W.: A criterion for A-stability of linear multistep integration formulae. Computing 3 (1968), 280–285.CrossRefGoogle Scholar
  9. 9.
    Liniger, W. and R.A. Willoughby: Efficient numerical integration methods for stiff system of differential equations. IBM Res. Report RC-1970, 1967.Google Scholar
  10. 10.
    Marden, M.: Geometry of Polynomials (2nd ed.). Amer. Mathem. Society, Providence, 1966.Google Scholar

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© Springer Basel AG 1974

Authors and Affiliations

  • H. Brunner
    • 1
  1. 1.HalifaxCanada

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