Dynamical
Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear
Differential Equations
H.C.
Yee and P.K.Sweby
RNR Technical Report
RNR-92-008 February 1992
Abstract
The
global asymptotic nonlinear behavior of 11 explicit and implicit
time discretizations for four 2 * 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.
To
view the full report:
Postscript
Version -254k
PDF
Version -221k
To
read this file you will need the free Adobe
Acrobat Reader.
Graphics
from this paper
|