Geometric proofs of numerical stability for delay equations

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 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 ∈ ℜ and b ∈ ∁. Furthermore, we prove that Radau‐IIA methods are stable for the subclass of equations where a = α + iγ with α, γ ∈ ℜ, γ sufficiently small, and b ∈ ∁.