The Stability Analysis of the $θ$-Methods for Delay Differential Equations (Journal of Computational Mathematics, 1996)
This paper deals with the stability analysis of $θ -$methods for the numerical solution of delay differential equations (DDEs). We focus on the behaviour of such methods in the solution of the linear test equation $y^{′}(t)=a(t)y(t)+b(t)y(t-τ )$, where $τ >0$, $a(t)$ and $b(t)$ are functions from $R$ to $C$. It is proved that the linear $θ -$method and the one-leg $θ -$method are TGP-stable if and only if $θ =1.$