Backward error analysis for multistep methods
E. Hairer
Abstract.
In recent years, much insight into the numerical solution
of ordinary differential equations by one-step methods
has been obtained with a backward error analysis. It allows one to
explain interesting phenomena such as the almost conservation of energy,
the linear error growth in Hamiltonian systems, and the existence
of periodic solutions and invariant tori. In the present
article, the formal backward error analysis as well
as rigorous, exponentially small error estimates are extended to
multistep methods. A further extension to partitioned
multistep methods is outlined,
and numerical illustrations of the theoretical results
are presented.
Key Words.
Multistep methods,
backward error analysis,
parasitic modified equation,
long-time itegration.