Geometric proofs of numerical stability for delay equations (Q2713138)

Asymptotic stability properties of implicit Runge-Kutta methods for delay-differential equations of the form NEWLINE\[NEWLINEy'(t)= ay(t)+ by(t-1),\quad a,b\in \mathbb{C},\tag{\(*\)}NEWLINE\]NEWLINE with initial condition \(y(t)= g(t)\), \(-1\leq t\leq 0\), where \(g\) is complex valued, is studied. It is shown that symmetric methods and all methods of even order cannot be unconditionally stable for \((*)\). However, many of them are stable for \((*)\), where \(a\in\mathbb{R}\) and \(b\in\mathbb{C}\). Further, it is proved that Radau -IIA methods are stable for \((*)\), where \(a= \alpha+ i\gamma\) with \(\alpha,\gamma\in \mathbb{R}\), \(\gamma\) sufficiently small and \(b\in\mathbb{C}\).