The stability of the \(\theta\)-methods in the numerical solution of delay differential equations with several delay terms (Q1899950)

The paper is concerned with the stability analysis of numerical methods for the solution of delay differential equations. The authors investigate the numerical stability of the linear \(\theta\)-method, the one-leg \(\theta\)-method and a new \(\theta\)-method. The concepts of \((\delta_1, \delta_2, \dots, \delta_m)\)-stability and of \(GP_m\)-stability are used when the numerical method is applied to the linear test delay equation with \(m\) delay terms. For \(m=2\) it is shown that the linear \(\theta\)-method and the new \(\theta\)-method are \(GP_2\)-stable if and only if \( {1\over 2}\leq \theta\leq 1\) and that the one-leg \(\theta\)-method is \(GP_2\)-stable if and only if \(\theta =1\). For \(m>2\) for the linear \(\theta\)-method the same \(GP_m\)-stability result is obtained. Numerical experiments from the last section illustrate the stability results.