Nonautonomous stability of linear multistep methods

• Published in: IMA journal of numerical analysis Vol. 30; no. 2; pp. 525 - 542
• Main Authors: Boutelje, B. R., Hill, A. T.
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.04.2010
• Subjects: Exact sciences and technology; Global analysis, analysis on manifolds; Initial value problems; Instability; linear multistep methods; Lyapunov functions; Mathematical analysis; Mathematical models; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Scalars; Sciences and techniques of general use; Stability; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Numerical analysis; Lyapunov functions; Multistep method; Initial value problem; linear multistep methods; Lipschitz function; Linear stability; Convex function; Numerical stability; stability; Lyapunov function
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/drn070

A linear scalar nonautonomous initial-value problem (IVP) is governed by a scalar λ(t) with a nonpositive real part. For a wide class of linear multistep methods, including BDF4–6, it is shown that negative real λ(t) may be chosen to generate instability in the method when applied to the IVP. However, a uniform-in-time stability result holds when λ(·) 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.