Stability in linear multistep methods for pure delay equations

The stability regions of linear multistep methods for pure delay equations are compared with the stability region of the delay equation itself. A criterion is derived stating when the numerical stability region contains the analytical stability region. This criterion yields an upper bound for the integration step (conditional Q-stability). These bounds are computed for the Adams-Bashforth, Adams-Moulton and backward differentiation methods of orders ?8. Furthermore, symmetric Adams methods are considered which are shown to be unconditionally Q-stable. Finally, the extended backward differentiation methods of Cash are analysed.