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Symmetric linear multistep. (English) Zbl 1149.65054
Summary: Some important early contributions of Germund Dahlquist are reviewed [see e. g. Math. Scand. 4, 33–53 (1956; Zbl 0071.11803)] and their impact to recent developments in the numerical solution of ordinary differential equations is shown. This work is an elaboration of a talk presented in the Dahlquist session at the SciCADE05 conference in Nagoya.
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE
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[2]G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand., 4 (1956), pp. 33–53.
[3]G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Trans. Royal Inst. Technol., vol. 130, Stockholm, Sweden, 1959.
[4]G. Dahlquist, 33 years of numerical instability, part I, BIT, 25 (1985), pp. 188–204. · Zbl 0579.65090 · doi:10.1007/BF01934997
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[6]E. Hairer, Backward error analysis for multistep methods, Numer. Math., 84 (1999), pp. 199–232. · Zbl 0941.65077 · doi:10.1007/s002110050469
[7]E. Hairer and C. Lubich, Symmetric multistep methods over long times, Numer. Math., 97 (2004), pp. 699–723. · Zbl 1060.65074 · doi:10.1007/s00211-004-0520-2
[8]E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn., Springer Series in Computational Mathematics 31, Springer, Berlin, 2006.
[9]P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons Inc., New York, 1962.
[10]J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl., 18 (1976), pp. 189–202. · Zbl 0359.65060 · doi:10.1093/imamat/18.2.189
[11]G. D. Quinlan and S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits, Astron. J., 100 (1990), pp. 1694–1700. · doi:10.1086/115629