The existence of stepsize-coefficients for boundedness of linear multistep methods

Abstract

In this paper we study linear multistep methods (LMMs) for the numerical solution of initial value problems. In the context of semidiscrete approximations of partial differential equations, much attention was paid in the literature to LMMs fulfilling special requirements indicated by the terms total-variation-diminishing, strong-stability-preserving and monotonicity. Stepsize restrictions, for the fulfillment of these requirements, were studied in many papers.These special requirements imply essential boundedness properties for the numerical methods, among which the property of being total-variation-bounded. Unfortunately, for many LMMs, the above requirements are violated, so that one cannot conclude via them that the methods are (total-variation) bounded.In this paper, we focus on stepsize restrictions for boundedness directly – rather than via the detour of the above special requirements. We present conditions by means of which one can check, for given LMMs, whether or not nontrivial stepsize restrictions exist guaranteeing boundedness.We illustrate the relevance of the above conditions by applying them to various classes of well-known LMMs, hereby supplementing earlier results, for these classes, given in the literature.