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Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential EquationsThe global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.
Document ID
19970027855
Acquisition Source
Ames Research Center
Document Type
Reprint (Version printed in journal)
Authors
Yee, H. C.
(NASA Ames Research Center Moffett Field, CA United States)
Sweby, P. K.
(Reading Univ. United Kingdom)
Date Acquired
August 17, 2013
Publication Date
January 1, 1995
Publication Information
Publication: Computer Fluid Dynamics
Publisher: Overseas Publishers Association
Volume: 4
Subject Category
Numerical Analysis
Report/Patent Number
NASA-TM-112909
NAS 1.15:112909
RNR-92-008
Meeting Information
Meeting: Numerical Methods in Fluid Mechanics
Location: Lausanne
Country: Switzerland
Start Date: September 25, 1991
End Date: September 27, 1991
Accession Number
97N26741
Distribution Limits
Public
Copyright
Public Use Permitted.
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