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Asymptotic expansions and backward analysis for numerical integrators

For numerical integrators of ordinary differential equations we compare the theory of asymptotic expansions of the global error with backward error analysis. On a formal level both approaches are equivalent. If, however, the arising divergent series are truncated, important features such as the semigroup property, structure perservation and exponentially small estimates over long times are valid only for the backward error analysis. We consider one-step methods as well as multistep methods, and we illustrate the theoretical results on several examples. In particular, we study the preservation of weakly stable limit cycles by symmetric methods. HAIRER, Ernst, LUBICH, Christian. Asymptotic expansions and backward analysis for numerical integrators. In: Rafael de la Llave; Linda R. Petzold & Jens Lorenz. Dynamics of algorithms. New York : Springer, 2000. p. 91-106