A collocation formulation of multistep methods for variable step-size extensions

Authors:
Carmen Arévalo;Claus Führer;Mónica Selva
Affiliations:
Department of Scientific Computing and Statistics, Simón Bolivar University, Apartado 89000, Caracas 1080-A, Venezuela;Numerical Analysis, Centre for Mathematical Sciences, Lund University, Box 118, S-221 00 Lund, Sweden;Center for Statistics and Mathematical Software (CESMA), Simón Bolivar University, Apartado 89000, Caracas 1080-A, Venezuela
Venue:
Applied Numerical Mathematics
Year:
2002

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Abstract

Multistep methods are classically constructed by specially designed difference operators on an equidistant time grid. To make them practically useful, they have to be implemented by varying the step-size according to some error-control algorithm. It is well known how to extend Adams and BDF formulas to a variable step-size formulation. In this paper we present a collocation approach to construct variable step-size formulas. We make use of piecewise polynomials to show that every k-step method of order k + 1 has a variable step-size polynomial collocation formulation.