Geometric proofs of numerical stability for delay equations

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 uncondi...

Full description

Saved in:
Bibliographic Details
Published in:IMA journal of numerical analysis 2001-01, Vol.21 (1), p.439-450
Main Authors: Guglielmi, Nicola, Hairer, Ernst
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Sciences and techniques of general use
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 ∈ ∁.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/21.1.439