Nonautonomous stability of linear multistep methods

Reference:

Boutelje, B. R. and Hill, A. T., 2010. Nonautonomous stability of linear multistep methods. IMA Journal of Numerical Analysis, 30 (2), pp. 525-542.

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Official URL:

http://dx.doi.org/10.1093/imanum/drn070

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.

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