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 ∈ ∁.

Geometric proofs of numerical stability for delay equations

GUGLIELMI N;
2001

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 ∈ ∁.
Stability of Runge-Kutta methods, delay differential equations, geometric proofs
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/1741
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