In this paper we provide an analysis of monotonicity properties for linear multistep methods. These monotonicity properties include positivity and the diminishing of total variation. We also pay particular attention to related boundedness properties such as the total-variation-bounded (TVB) property.In the analysis the multistep methods are considered in combination with suitable starting procedures. This allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results.

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Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Hundsdorfer, W., Ruuth, S. J., & Spiteri, R. J. (2002). Monotonicity-preserving linear multistep methods. Modelling, Analysis and Simulation [MAS]. CWI.