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

TL;DR

This paper derives exact algebraic values for the maximum step-size coefficients ensuring boundedness in linear multistep methods, improving understanding of stability properties for numerical solutions of differential equations.

Contribution

It introduces rigorous methods to verify sign conditions for step-size coefficients and determines exact optimal values for several multistep method families.

Findings

Exact algebraic values of SCBs for BDF methods (1-6 steps)

Exact algebraic values of SCBs for Adams–Bashforth methods (1-3 steps)

Confirmation of existence or non-existence of positive SCBs in studied families

Abstract

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). A 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 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 $1\le k\le 6$ steps) and in the Adams--Bashforth family (with $1\le k\le 3$ steps).