Asymptotic expansions and backward analysis for numerical
integrators
E. Hairer
and
Ch. Lubich
Abstract.
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.
Key Words.
Asymptotic expansions, backward error analysis,
one-step methods, multistep methods, long-time behavior.