Special boundedness properties in numerical initial value problems
KAUST Grant NumberFIC/2010/05
Online Publication Date2011-09-21
Print Publication Date2011-12
Permanent link to this recordhttp://hdl.handle.net/10754/599687
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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.
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.
SponsorsWe 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).
JournalBIT Numerical Mathematics