Advanced Search
Article Contents
Article Contents

Regarding the absolute stability of Størmer-Cowell methods

Abstract Related Papers Cited by
  • High order variants of the classical Størmer-Cowell methods are still a popular class of methods for computations in celestial mechanics. In this work we shall investigate the absolute stability of Størmer-Cowell methods close to zero, and present a characterization of the stability of methods of all orders. In particular, we show that many methods are not absolutely stable at any point in a neighborhood of the origin.
    Mathematics Subject Classification: Primary: 65L06, 70F15; Secondary: 65L20, 70M20.


    \begin{equation} \\ \end{equation}
  • [1]

    G. Dahlquist, On accuracy and unconditional stability of linear multistep methods for second order differential equations, BIT; Nordisk Tidskrift for Informationsbehandling (BIT), 18 (1978), 133-136.doi: 10.1007/BF01931689.


    W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numerische Mathematik, 3 (1961), 381-397.doi: 10.1007/BF01386037.


    K. Grazier, W. Newman, J. Hyman, P. Sharp and D. GoldsteinAchieving Brouwer's law with high-order Störmer multistep methods, ANZIAM J., 46 (2004/05), C786–-C804.


    E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numerica, 12 (2003), 399-450.doi: 10.1017/S0962492902000144.


    E. Hairer, S. Nørsett and G. Wanner, "Solving Ordinary Differential Equations: Nonstiff Problems, vol. 1," Springer Verlag, 1993.


    E. Hairer and G. Wanner, "Solving Ordinary Differential Equations {II}: Stiff and Differential-Algebraic Problems, vol. 2," Springer, 2004.


    P. Henrici, "Discrete Variable Methods in Ordinary Differential Equations, vol. 1," New York: Wiley, 1962.


    J. Lambert, "Computational Methods in Ordinary Differential Equations," Wiley New York, 1973.


    J. Lambert and I. Watson, Symmetric multistip methods for periodic initial value problems, IMA Journal of Applied Mathematics, 18 (1976), 189-202.doi: 10.1093/imamat/18.2.189.


    W. I. Newman, F. Varadi, A. Y. Lee, W. M. Kaula, K. R. Grazier and J. M. Hyman, Numerical integration, Lyapunov exponents and the outer Solar System, Bulletin of the American Astronomical Society, 32 (2000), 859.


    G. Quinlan and S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits, The Astronomical Journal, 100 (1990), 1694-1700.


    P. Sharp, Comparisons of high order stormer and explicit Runge-kutta Nyström methods for N-body simulations of the solar system, Tech. Rep., Department of Mathematics, The University of Auckland, New Zealand, (2000).


    E. Stiefel and D. G. Bettis, Stabilization of Cowell's method, Numerische Mathematik, 13 (1969), 154-175.doi: 10.1007/BF02163234.


    E. Thorbergsen, "Undersøkelse av Noen Metoder for Baneproblemer," Master's thesis, Norges Tekniske Høyskole(NTH), Trondheim, Norway, 1976.


    F. Varadi and B. Runnegar, Successive refinements in long-term integrations of planetary orbits, The Astrophysical Journal, 592 (2003), 620-630.

  • 加载中

Article Metrics

HTML views() PDF downloads(81) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint