Nonautonomous stability of linear multistep methods

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.