Stability analysis of \(\theta\)-methods for neutral functional- differential equations
type: Item
instance of: scholarly article
title: Stability analysis of \(\theta\)-methods for neutral functional- differential equations
zbMATH Open document ID: 0824.65081
author: Yunkang Liu
published in: Numerische Mathematik
publication date: 1995-11-13T00:00:00Z
review text: This paper deals with the numerical stability of the neutral functional differential equation \(y'(t) = ay(t) + by(qt) + cy'(pt)\), \(t > 0\). It is proved that \(\theta\)-methods are convergent if \(| c| < 1\). Experiments suggest they are divergent if \(| \theta|\) is large. The problem is transferred to a neutral equation with constant time lags. Using the later equation as a test model, it is shown that the linear \(\theta\)-method is \(\Lambda\)-stable if \(\text{Re }a < 0\) and \(| a| > | b|\) if and only if \(\theta \geq 1/2\) and the one-leg \(\theta\)-method is \(\Lambda\)-stable if \(\theta = 1\). It is also shown that inappropriate stepsize causes spurious solutions in the marginal case when \(\text{Re }a < 0\) and \(| a| = | b|\).
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 34K05
zbMATH DE Number: 770169
zbMATH Keywords: theta method
zbMATH Keywords: convergence
zbMATH Keywords: stepsize control
zbMATH Keywords: numerical stability
zbMATH Keywords: neutral functional differential equation
reviewed by: Stephen L. Campbell
MaRDI profile type: Publication
full work available at URL: s002110050129
OpenAlex ID: W1969409177
DOI: 10.1007/S002110050129
description: scientific article; zbMATH DE number 770169