Special boundedness properties in numerical initial value problems

W. Hundsdorfer, A. Mozartova, M. N. Spijker

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


For Runge-Kutta methods, linear multistep methods and other classes of general linear methods much attention has been paid in the literature to important nonlinear stability properties known as total-variation-diminishing (TVD), strong stability preserving (SSP) and monotonicity. Stepsize conditions guaranteeing these properties were studied by Shu and Osher (J. Comput. Phys. 77:439-471, 1988) and in numerous subsequent papers. Unfortunately, for many useful methods it has turned out that these properties do not hold. For this reason attention has been paid in the recent literature to the related and more general properties called total-variation-bounded (TVB) and boundedness. In the present paper we focus on stepsize conditions guaranteeing boundedness properties of a special type. These boundedness properties are optimal, and distinguish themselves also from earlier boundedness results by being relevant to sublinear functionals, discrete maximum principles and preservation of nonnegativity. Moreover, the corresponding stepsize conditions are more easily verified in practical situations than the conditions for general boundedness given thus far in the literature. The theoretical results are illustrated by application to the two-step Adams-Bashforth method and a class of two-stage multistep methods. © 2011 Springer Science + Business Media B.V.
Original languageEnglish (US)
Pages (from-to)909-936
Number of pages28
JournalBIT Numerical Mathematics
Issue number4
StatePublished - Sep 21 2011
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): FIC/2010/05
Acknowledgements: We thank the referee for comments which have resulted in an improved presenta-tion of our work. The work of A. Mozartova is supported by a grant from the Netherlands Organisationfor Scientific Research NWO. The work of W. Hundsdorfer for this publication was partially supported byAward No. FIC/2010/05 from King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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