Characteristic Roots of Discretized Functional Differential Equations

If an ordinary differential equation is solved by using a linear multistep method, then it is well-known that the asymptotic behaviour of the error is determined by the characteristic roots of the discretized differential equation. In particular the so-called essential roots (the roots on or close to the unit circle) play an important role. If a functional differential equation is solved by means of a linear multistep method, then the number of characteristic roots of the discretized equation is inversely proportional to the stepsize used. In this paper it is shown that only a fixed number of these roots are strictly inside the unit circle, and all the others are “essential”.