Nonlinear stability of general linear methods for neutral delay differential equations
type: Item
instance of: scholarly article
title: Nonlinear stability of general linear methods for neutral delay differential equations
zbMATH Open document ID: 1167.65046
author: Wansheng Wang
author: Kai Su
author: Shoufu Li
published in: Journal of Computational and Applied Mathematics
publication date: 2009-02-25T00:00:00Z
review text: The authors consider the numerical solutions of neutral delay differential equations in the general nonlinear form, \[ \begin{cases} y'(t) = f(t,y(t), y(t-\tau), y'(t-\tau)), & t\geq 0,\\ y(t) = \phi(t), & t\leq 0. \end{cases} \] The main purpose is to study the stability of general linear methods, such as backward differentiation formulae (BDF's), extended and modified extended BDF's, parallel multi-value hybrid methods, Runge-Kutta, multi-step Runge-Kutta. A general class of methods to which the results can be applied is described in section 3 of the paper, while in section 2 the main theoretical results attending to some required properties of the components of the differential equation are given. Section 3 also contains many definitions and comments related to stability that are enlarged with the introduction of the new concepts of GS(p), GAS(p), and weak GAS(p) stability of general linear methods with linear interpolation. Many implications between these and other concepts of stability, as well as the relation with \((k,p,0)\)-algebraic stability are investigated, specially in section 4, where the main results and proofs of the paper are given. The paper ends with some examples of implementation of multi-step Runge-Kutta methods and some numerical experiments to support the results of the work.
reviewed by: Josep J. Masdemont
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34K50
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 65L06
zbMATH DE Number: 5518722
zbMATH Keywords: nonlinear neutral delay differential equations
zbMATH Keywords: general linear methods
zbMATH Keywords: numerical stability
zbMATH Keywords: algebraic stability
zbMATH Keywords: backward differentiation formulae
zbMATH Keywords: multi-step Runge-Kutta methods
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: j.cam.2008.05.050
OpenAlex ID: W2041676064
cites work: On the contractivity and asymptotic stability of systems of delay differential equations of neutral type
cites work: Strong contractivity properties of numerical methods for ordinary and delay differential equations
cites work: Numerical stability of nonlinear delay differential equations of neutral type
cites work: On error growth functions of Runge-Kutta methods
cites work: Q3860165
cites work: Stability analysis of numerical methods for systems of neutral delay-differential equations
cites work: Stability analysis of Runge-Kutta methods for nonlinear delay differential equations
cites work: Linear stability of general linear methods for systems of neutral delay differential equations
cites work: Nonlinear stability of general linear methods for delay differential equations
cites work: Q4229393
cites work: The asymptotic stability of one-parameter methods for neutral differential equations
cites work: Stability and $B$-convergence properties of multistep Runge-Kutta methods
cites work: The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations
cites work: The stability of linear multistep methods for linear systems of neutral differential equations
cites work: A stable numerical approach for implicit non-linear neutral delay differential equations
cites work: Q5866785
cites work: Nonlinear stability of one-leg methods for delay differential equations of neutral type
cites work: Nonlinear stability of Runge-Kutta methods for neutral delay differential equations
cites work: Q4795619
cites work: NGP(\(\alpha\))-stability of general linear methods for NDDEs
DOI: 10.1016/J.CAM.2008.05.050
description: scientific article; zbMATH DE number 5518722