Numerische Mathematik

, Volume 76, Issue 4, pp 441–462 | Cite as

The life-span of backward error analysis for numerical integrators

  • E. Hairer
  • C. Lubich


Backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations. The numerical solution is formally interpreted as the exact solution of a perturbed differential equation, given as a formal and usually divergent series in powers of the step size. For a rigorous analysis, this series has to be truncated. In this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation. Results on the long-time behaviour of numerical solutions are obtained in this way. We present applications to the numerical phase portrait near hyperbolic equilibrium points, to asymptotically stable periodic orbits and Hopf bifurcation, and to energy conservation and approximation of invariant tori in Hamiltonian systems.

Mathematics Subject Classification (1991):65L05, 65L70 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • E. Hairer
    • 1
  • C. Lubich
    • 2
  1. 1. Département de Mathématiques, Université de Genève, CH-1211 Genève 24, Switzerland; e-mail: CH
  2. 2. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany; e-mail: DE

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