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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 dynamicsSpurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.
Document ID
19920034802
Acquisition Source
Legacy CDMS
Document Type
Reprint (Version printed in journal)
Authors
Yee, H. C.
(NASA Ames Research Center Moffett Field, CA, United States)
Sweby, P. K.
(Reading, University United Kingdom)
Griffiths, D. F.
(Dundee, University United Kingdom)
Date Acquired
August 15, 2013
Publication Date
December 1, 1991
Publication Information
Publication: Journal of Computational Physics
Volume: 97
ISSN: 0021-9991
Subject Category
Numerical Analysis
Accession Number
92A17426
Distribution Limits
Public
Copyright
Other

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