Backward error analysis for multistep methods

In recent years, much insight into the numerical solution of ordinary diffe rential equations by one-step methods has been obtained with a backward err or analysis. It allows one to explain interesting phenomena such as the alm ost 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, exponenti ally small error estimates are extended to multistep methods. A further ext ension to partitioned multistep methods is outlined, and numerical illustra tions of the theoretical results are presented.