Q1181869
type: Item
instance of: scholarly article
title: Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I: The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics
zbMATH Open document ID: 0760.65087
DOI: 10.1016/0021-9991(91)90001-2
author: H. C. Yee
author: David F. Griffiths
author: Peter K. Sweby
published in: Journal of Computational Physics
publication date: 1992-06-27T00:00:00Z
full work available at URL: https://digitalcommons.unl.edu/nasapub/280
review text: This paper is part I of a series of research papers on a dynamical approach to study numerical methods for nonlinear differential equations and it serves as an introduction to the parts II and III. About 21 pages are devoted to the introduction, motivation and relevance of this approach, and the implications, recommendations and conclusions. On 30 pages it is demonstrated how differently differential equations and their discretized counterparts can behave. Numerical experiments for a scalar nonlinear ordinary differential equation \(u'=\alpha S(u)\) with \(S(u)=u(1-u)\) (the logistic equation) or \(S(u)=u(1-u)(b-u)\) are pictured in terms of fixpoint diagrams and bifurcation diagrams. Several two level and three level methods are tested. One of the findings is that spurious steady-state solutions can occur below the linearized stability limit. A rough outlook for partial differential equations is given on 9 pages. I close by citing the authors: ``Thus the mission of this paper is not to provide the answer or theory or to illustrate the connection of dynamical behavior of practical partial differential equations to their discretized counterpart, but rather to gain insight into the nonlinear features unconventional to this type of study and concentrate on the fundamentals. In order to bring out the new features, the illustrations concentrate on simple scalar differential equation examples in which the exact solutions of the differential equations are known.''.
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 29001
zbMATH Keywords: time discretization
zbMATH Keywords: algorithm
zbMATH Keywords: dynamics of numerics
zbMATH Keywords: stability of methods
zbMATH Keywords: numerical experiments
zbMATH Keywords: nonlinear differential equations
zbMATH Keywords: logistic equation
zbMATH Keywords: fixpoint diagrams
zbMATH Keywords: bifurcation diagrams
zbMATH Keywords: spurious steady-state solutions
reviewed by: Walter Zulehner
describes a project that uses: AUTO-86
describes a project that uses: AUTO
MaRDI profile type: Publication
OpenAlex ID: W1991270927
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