The Numerical Stability of Linear Multistep Methods for Delay Differential Equations with Many Delays

This paper deals with the stability analysis of linear multistep methods for the numerical solution of delay differential equations (DDEs) with many delays. We focus on the stability behaviour of such methods when they are applied to a scalar, linear DDE, with many constant delays and complex coefficients. The concept of GPm-stability is introduced. Under a sufficient condition for the test equation to be stable, the linear multistep method, when combined with a certain Lagrange interpolation, must be stable for all stepsizes. It is proven that a linear multistep method is GPm-stable if and only if it is A-stable for ordinary differential equations (ODEs).