The Stability of Linear Multistep Methods for Linear Systems of Neutral Differential Equations

Abstract

This paper deals with the numerical solution of initial value problems for systems of neutral differential equations $$y'(t)=f(t,y(t),y(t- \tau ),y'(t- \tau )), t ＞ 0, $$ $$y(t) = φ(t) \ t＜0,$$ where $\tau＞ 0, f$ and φ denote given vector-valued functions. The numerical stability of a linear multistep method is investigated by analysing the solution of the test equations $y'(t)=Ay(t) + By(t-\tau) + Cy'(t-\tau),$ where $A, B$ and $C$ denote constant complex $N \times N$-matrices, and $\tau ＞ 0$. We investigate the properties of adaptation of the linear multistep method and the characterization of the stability region. It is proved that the linear multistep method is NGP-stable if and only if it is A-stable for ordinary differential equations.