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We consider numerical methods for solving ordinary differential equations. Asymptotic expansions of the global error (Henrici 1962, Gragg 1964, Stetter 1973, ...) on the one hand, and backward error analysis (Feng Kang 1991, Sanz-Serna 1992, Yoshida 1993, ...) on the other hand, both have as objective a better understanding of the global error of integration methods.
On a formal level both approaches yield equivalent representations of the numerical solution, but as soon as rigorous estimates are desired, the results are completely different. The reason is that in the theory of asymptotic expansions the diverging series are truncated in the space of solutions, whereas in the theory of backward error analysis the series are truncated in the space of vector fields. Due to its good approximation properties, backward error analysis gives new insight into numerical methods for a variety of problems: long-time integration of Hamiltonian or reversible differential equations (conservation of energy and of KAM tori), asymptotically stable periodic orbits, Hopf bifurcation, invariant tori of dissipatively perturbed Hamiltonian systems. The theoretical results will be illustrated with several examples.
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