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IMA Journal of Numerical Analysis Advance Access published online on February 20, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn070
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Nonautonomous stability of linear multistep methods

B. R. Boutelje and A. T. Hill{dagger}

Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK

{dagger} Corresponding author. Email: ath{at}maths.bath.ac.uk

Received on 7 April 2008. Revised on 19 September 2008.


  Abstract

A linear scalar nonautonomous initial-value problem (IVP) is governed by a scalar {lambda}(t) with a nonpositive real part. For a wide class of linear multistep methods, including BDF4–6, it is shown that negative real {lambda}(t) may be chosen to generate instability in the method when applied to the IVP. However, a uniform-in-time stability result holds when {lambda}(·) is a Lipschitz function, subject to a related restriction on h. The proof involves the construction of a Lyapunov function based on a convex combination of G-norms.

Key Words: linear multistep methods; stability; Lyapunov functions


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