Geometric proofs of numerical stability for delay equations
Nicola Guglielmi and Ernst Hairer
Abstract.
In this paper, asymptotic stability properties of
implicit Runge Kutta methods
for delay differential equations are considered
with respect to the test equation
$y'(t) = a\, y(t) + b\, y(t\,-1)$ with $a,b \in \C$.
In particular, we prove that symmetric methods
and all methods of even order
cannot be unconditionally stable with respect
to the considered test equation, while many of them are stable on problems
where $a \in \R$ and $b\in\C$.
Furthermore, we prove that Radau~IIA methods are stable on the subclass
for equations where $a = \alpha + i \gamma$ with $\alpha, \gamma \in \R$,
$\gamma$ sufficiently small, and $b\in \C$.
Key Words.
Dealay differential equations, Runge-Kutta methods,
asymptotic stability, root locus technique.