The stability problem for linear multistep methods: Old and new results

The paper reviews results on rigorous proofs for stability properties of classes of linear multistep methods (LMMs) used either as IVMs or as BVMs. The considered classes are not only the well-known classical ones (BDF, Adams, ...) along with their BVM correspondent, but also those which were considered unstable as IVMs, but stable as BVMs. Among the latter we find two classes which deserve attention because of their peculiarity: the TOMs (top order methods) which have the highest order allowed to a LMM and the Bs-LMMs which have the property to carry with each method its natural continuous extension.