Lecturer: Lajos Lóczi (ELTE IK Department of Numerical Analysis, BME Department of Differential Equations)
Abstract: Monotonicity or boundedness properties (e.g. strong-stability-preserving, total-variation-diminishing or total-variation-boundedness properties) for linear multistep methods (LMMs) can be guaranteed by imposing step-size restrictions on the methods. To describe these restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) and its generalization, the step-size coefficient for boundedness (SCB coefficient). A LMM with larger SCM or SCB is more efficient. The computation of the maximum SCM for a particular LMM is straightforward, however, it is more challenging to decide whether a SCB exists, or determine if a given number is a SCB. Based on some recent theorems in the literature we present methods to find the exact optimal SCB for a LMM. As an illustration, we consider SCBs in the BDF, extrapolated BDF, and Adams--Bashforth families.