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W. Auzinger, F. Kramer:
"On the stability and error structure of BDF schemes applied to sectorial problems";
Talk: Time Integration of Evolution Equations, Innsbruck; 2007-09-12 - 2007-09-15.

We consider BDF schemes applied to linear evolution equations u'=Au+f and study their stability and damping behavior. The relevant A(α)-stable BDF methods are `uniformly strongly stable'; in particular, explicit inclusion sets for the parasitic roots of the characteristic polynomial ρ(ζ)-μ σ(ζ) have recently been specified which are valid for any μ in the stability domain. This can be used to extend the work of former authors by deriving quantitative stability estimates for sectorial operators A.

The two-step BDF scheme is considered in particular detail, and the structure of its global error is described by an asymptotic expansion with a precise, uniform estimate for its remainder. This expansion involves non-smooth perturbations which rapidly decay for t>0 due to the damping properties of the coefficients of the discrete resolvent.

BDF methods, stability, asymptotic error expansion

http://publik.tuwien.ac.at/files/pub-tm_5797.pdf