Linear multistep methods (LMMs) applied to approximate the solution of initial value problems—typically arising from method-of-lines semidiscretizations of partial differential equations—are often required to have certain monotonicity or boundedness properties (e.g., strong-stability-preserving, total-variation-diminishing or total-variation-boundedness properties). These properties can be guaranteed by imposing step-size restrictions on the methods. To qualitatively describe the step-size restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) or its generalization, the step-size coefficient for boundedness (SCB). An LMM with larger SCM or SCB is more efficient, and the computation of the maximum SCM for a particular LMM is now straightforward. However, it is more challenging to decide whether a positive SCB exists, or determine if a given positive number is a SCB. Theorems involving sign conditions on certain linear recursions associated to the LMM have been proposed in the literature that allow us to answer the above questions: the difficulty with these theorems is that there are in general infinitely many sign conditions to be verified. In this work, we present methods to rigorously check the sign conditions. As an illustration, we confirm some recent numerical investigations concerning the existence of positive SCBs in the BDF and in the extrapolated BDF (EBDF) families. As a stronger result, we determine the optimal values of the SCBs as exact algebraic numbers in the BDF family (with 1 ≤ k ≤ 6 steps) and in the Adams–Bashforth family (with 1 ≤ k ≤ 3 steps).
Linear multistep methods Strong stability preservation Step-size coefficient for monotonicity Step-size coefficient for boundedness
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Ouaknine, J., Worrell, J.: Decision problems for linear recurrence sequences. In: reachability problems, 6th International Workshop, RP 2012, Bordeaux, France, September 17–19 (2012). doi:10.1007/978-3-642-33512-9_3Google Scholar
Ouaknine, J., Worrell, J.: Positivity problems for low-order linear recurrence sequences. In: proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (2014). doi:10.1137/1.9781611973402.27Google Scholar
Ouaknine, J., Worrell, J.: Automata, languages, and programming. Part II: on the positivity problem for simple linear recurrence Sequences, pp. 318–329. Springer, Heidelberg (2014)Google Scholar
Ouaknine, J., Worrell, J.: Ultimate positivity is decidable for simple linear recurrence sequences, pp. 330–341. Springer, Heidelberg (2014)MATHGoogle Scholar