Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type - Part II: Linear multistep methods

Abstract

It is shown that appropriate linear multi-step methods (LMMs) applied to singularly perturbed systems of ODEs preserve the geometric properties of the underlying ODE. If the ODE admits an attractive invariant manifold so does the LMM. The continuous as well as the discrete dynamical system restricted to their invariant manifolds are no longer stiff and the dynamics of the full systems is essentially described by the dynamics of the systems reduced to the manifolds. These results may be used to transfer properties of the reduced system to the full system. As an example global error bounds of LMM-approximations to singularly perturbed ODEs are derived.