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March  2014, 34(3): 1131-1146. doi: 10.3934/dcds.2014.34.1131

Regarding the absolute stability of Størmer-Cowell methods

Syvert P. Nørsett 1, and  Andreas Asheim 2,

1. 

Dept. of Mathematical Sciences, NTNU Trondheim, N-7491 Trondheim, Norway
2. 

Dept. Computer Science, University of Leuven, Belgium, BE-3001 Heverlee

Received  September 2012 Revised  October 2012 Published  August 2013

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.
Keywords: absolute stability., Multistep methods for second order problems, Størmer-Cowell methods.
Mathematics Subject Classification: Primary: 65L06, 70F15; Secondary: 65L20, 70M2.
Citation: Syvert P. Nørsett, Andreas Asheim. Regarding the absolute stability of Størmer-Cowell methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1131-1146. doi: 10.3934/dcds.2014.34.1131
References:
[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.  Google Scholar

[2]

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

[3]

K. Grazier, W. Newman, J. Hyman, P. Sharp and D. Goldstein, Achieving Brouwer's law with high-order Störmer multistep methods,, ANZIAM J., 46 ().   Google Scholar

[4]

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.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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.  Google Scholar

[10]

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. Google Scholar

[11]

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

[12]

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). Google Scholar

[13]

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

[14]

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

[15]

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

show all references

References:
[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.  Google Scholar

[2]

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

[3]

K. Grazier, W. Newman, J. Hyman, P. Sharp and D. Goldstein, Achieving Brouwer's law with high-order Störmer multistep methods,, ANZIAM J., 46 ().   Google Scholar

[4]

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.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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.  Google Scholar

[10]

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. Google Scholar

[11]

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

[12]

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). Google Scholar

[13]

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

[14]

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

[15]

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

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