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Linear multistep methods (LMMs) applied to approximate the solution ofinitial value problems---typically arising from method-of-linessemidiscretizations of partial differential equations---are often required tohave certain monotonicity or boundedness properties (e.g.strong-stability-preserving, total-variation-diminishing ortotal-variation-boundedness properties). These properties can be guaranteed byimposing step-size restrictions on the methods. To qualitatively describe thestep-size restrictions, one introduces the concept of step-size coefficient formonotonicity (SCM, also referred to as the strong-stability-preserving (SSP)coefficient) or its generalization, the step-size coefficient for boundedness(SCB). A LMM with larger SCM or SCB is more efficient, and the computation ofthe maximum SCM for a particular LMM is now straightforward. However, it ismore challenging to decide whether a positive SCB exists, or determine if agiven positive number is a SCB. Theorems involving sign conditions on certainlinear recursions associated to the LMM have been proposed in the literaturethat allow us to answer the above questions: the difficulty with these theoremsis that there are in general infinitely many sign conditions to be verified. Inthis work we present methods to rigorously check the sign conditions. As anillustration, we confirm some recent numerical investigations concerning theexistence of SCBs in the BDF and in the extrapolated BDF (EBDF) families. As astronger result, we determine the optimal values of the SCBs as exact algebraicnumbers in the BDF family (with $1\le k\le 6$ steps) and in theAdams--Bashforth family (with $1\le k\le 3$ steps).

Exact optimal values of step-size coefficients for boundedness of linear multistep methods