Special boundedness properties in numerical initial value problems

Handle URI:
http://hdl.handle.net/10754/599687
Title:
Special boundedness properties in numerical initial value problems
Authors:
Hundsdorfer, W.; Mozartova, A.; Spijker, M. N.
Abstract:
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.
Citation:
Hundsdorfer W, Mozartova A, Spijker MN (2011) Special boundedness properties in numerical initial value problems. Bit Numer Math 51: 909–936. Available: http://dx.doi.org/10.1007/s10543-011-0349-x.
Publisher:
Springer Nature
Journal:
BIT Numerical Mathematics
KAUST Grant Number:
FIC/2010/05
Issue Date:
21-Sep-2011
DOI:
10.1007/s10543-011-0349-x
Type:
Article
ISSN:
0006-3835; 1572-9125
Sponsors:
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).
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorHundsdorfer, W.en
dc.contributor.authorMozartova, A.en
dc.contributor.authorSpijker, M. N.en
dc.date.accessioned2016-02-28T06:07:33Zen
dc.date.available2016-02-28T06:07:33Zen
dc.date.issued2011-09-21en
dc.identifier.citationHundsdorfer W, Mozartova A, Spijker MN (2011) Special boundedness properties in numerical initial value problems. Bit Numer Math 51: 909–936. Available: http://dx.doi.org/10.1007/s10543-011-0349-x.en
dc.identifier.issn0006-3835en
dc.identifier.issn1572-9125en
dc.identifier.doi10.1007/s10543-011-0349-xen
dc.identifier.urihttp://hdl.handle.net/10754/599687en
dc.description.abstractFor 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.en
dc.description.sponsorshipWe 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).en
dc.publisherSpringer Natureen
dc.subjectBoundednessen
dc.subjectGeneral linear method (GLM)en
dc.subjectInitial value problemen
dc.subjectMethod of lines (MOL)en
dc.subjectMonotonicityen
dc.subjectOrdinary differential equation (ODE)en
dc.subjectStrong-stability-preserving (SSP)en
dc.subjectTotal-variation-bounded (TVB)en
dc.subjectTotal-variation-diminishing (TVD)en
dc.titleSpecial boundedness properties in numerical initial value problemsen
dc.typeArticleen
dc.identifier.journalBIT Numerical Mathematicsen
dc.contributor.institutionCentrum voor Wiskunde en Informatica, Amsterdam, Netherlandsen
dc.contributor.institutionLeiden University, Leiden, Netherlandsen
kaust.grant.numberFIC/2010/05en
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