The stability of the θ-methods in the numerical solution of delay differential equations with several delay terms

This paper deals with the stability analysis of numerical methods for the solution of delay differential equations (DDEs). We focus on the stability behaviour of the θ-methods in the solution of the following linear test equation with m delay terms: y′(t)=ay(t)+∑j=1mbjy(t−τ), t⩾0, y(t) = ø(t), t ⩽ 0, where a,bj (j = 1,2,…,m) ∈ C, τm ⩾ τm−1 ⩾ h. ⩾ τ1>0, and ø(t) is continuous and complex valued. For m = 2, it is shown that the linear θ-method and the new θ-method are GP2-stable if and only if 12 ⩽ θ ⩽ 1 and that the one-leg θ-method is GP2-stable if and only if θ = 1. In addition, for m > 2, we investigate the stability properties of the θ-methods with respect to the linear test equation and arrive at the same results.