Regular and singular [formula omitted]-blocking of difference corrected multistep methods for nonstiff index-2 DAEs

Abstract

There are several approaches to using nonstiff implicit linear multistep methods for solving certain classes of semi-explicit index 2 DAEs. Using β-blocked discretizations (Arévalo et al., 1996) Adams–Moulton methods up to order 4 and difference corrected BDF (Söderlind, 1989) methods up to order 7 can be stabilized. As no extra matrix computations are required, this approach is an alternative to projection methods. Here we examine some variants of β-blocking. We interpret earlier results as regular β-blocking and then develop singular β-blocking. In this nongeneric case the stabilized formula is explicit, although the discretization of the DAE as a whole is implicit. We investigate which methods can be stabilized in a broad class of implicit methods based on the BDF ρ polynomials. The class contains the BDF, Adams–Moulton and difference corrected BDF methods as well as other high order methods with small error constants. The stabilizing difference operator τ is selected by a minimax criterion for the moduli of the zeros of σ+τ. The class of explicit methods suitable as β-blocked methods is investigated. With singular β-blocking, Adams–Moulton methods up to order 7 can be stabilized with the stabilized method corresponding to the Adams–Bashforth methods.