Unified approach to spurious solutions introduced by time discretization Part II: BDF-like methods

Abstract

It has been proved inter alia in part I of the present paper (Iserles et al., 1991) that irreducible multistep methods for ordinary differential equations may possess period-2 solutions as asymptotic states if and only if σ(−1)≠0, where the underlying method is ∑^m_k=0ρκyn+k = h ∑^m_k=0^σκf(yn+k) and σ(z):=∑^m_k=0^σκ^z^k. We provide an alternative proof of that statement and examine in detail properties of methods that obey σ(−1)=0. By using a variation of the original proof of the first Dahlquist barrier (Henrici, 1962), we establish an attainable upper bound on the order of zero-stable multistep methods with the aforementioned feature. Moreover, we modify the concept of backward differentiation formulae (BDF) to require that σ(−1)=0. A zero-stability bound on the ensuing methods is produced by extending the method of proof in (Hairer & Wanner, 1983).