5,511 research outputs found
On spurious steady-state solutions of explicit Runge-Kutta schemes
The bifurcation diagram associated with the logistic equation v sup n+1 = av sup n (1-v sup n) is by now well known, as is its equivalence to solving the ordinary differential equation u prime = alpha u (1-u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. Runge-Kutta schemes applied to both the equation u prime = alpha u (1-u) and the cubic equation u prime = alpha u (1-u)(b-u) were studied computationally and analytically and their behavior was contrasted with the explicit Euler scheme. Their spurious fixed points and periodic orbits were noted. In particular, it was observed that these may appear below the linearized stability limits of the scheme and, consequently, computation may lead to erroneous results
Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics
Spurious 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
On Spurious Behavior of Super-Stable Implicit Methods
Th e objective of this paper is to gain a better understanding of the asymptotic nonlinear behavior of super-stable implicit linear multistep methods for constant time steps. Examples of the nonlinear effect caused by grid adaptation and super-stable implicit total variation diminishing (TVD) schemes on the overall performance of the numerical procedure are given. A method to minimize spurious steady-state numerical solutions is discussed
Global Asymptotic Behavior of Iterative Implicit Schemes
The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics
Pair Analysis of Field Galaxies from the Red-Sequence Cluster Survey
We study the evolution of the number of close companions of similar
luminosities per galaxy (Nc) by choosing a volume-limited subset of the
photometric redshift catalog from the Red-Sequence Cluster Survey (RCS-1). The
sample contains over 157,000 objects with a moderate redshift range of 0.25 < z
< 0.8 and absolute magnitude in Rc (M_Rc) < -20. This is the largest sample
used for pair evolution analysis, providing data over 9 redshift bins with
about 17,500 galaxies in each. After applying incompleteness and projection
corrections, Nc shows a clear evolution with redshift. The Nc value for the
whole sample grows with redshift as (1+z)^m, where m = 2.83 +/- 0.33 in good
agreement with N-body simulations in a LCDM cosmology. We also separate the
sample into two different absolute magnitude bins: -25 < M_Rc < -21 and -21 <
M_Rc < -20, and find that the brighter the absolute magnitude, the smaller the
m value. Furthermore, we study the evolution of the pair fraction for different
projected separation bins and different luminosities. We find that the m value
becomes smaller for larger separation, and the pair fraction for the fainter
luminosity bin has stronger evolution. We derive the major merger remnant
fraction f_rem = 0.06, which implies that about 6% of galaxies with -25 < M_Rc
< -20 have undergone major mergers since z = 0.8.Comment: ApJ, in pres
Star Formation in Cluster Galaxies at 0.2<z<0.55
The rest frame equivalent width of the [OII]3727 emission line, W(OII), has
been measured for cluster and field galaxies in the CNOC redshift survey of
rich clusters at 0.2<z<0.55. Emission lines of any strength in cluster galaxies
at all distances from the cluster centre, out to 2R_{200}, are less common than
in field galaxies. The mean W(OII) in cluster galaxies more luminous than
M_r^k<-18.5 + 5\log h (q_o=0.1) is 3.8 \pm 0.3 A (where the uncertainty is the
1 sigma error in the mean), significantly less than the field galaxy mean of
11.2 \pm 0.3 A. For the innermost cluster members (R<0.3R_{200}), the mean
W(OII) is only 0.3 \pm 0.4 A. Thus, it appears that neither the infall process
nor internal tides in the cluster induce detectable excess star formation in
cluster galaxies relative to the field. The colour-radius relation of the
sample is unable to fully account for the lack of cluster galaxies with
W(OII)>10 A, as expected in a model of cluster formation in which star
formation is truncated upon infall. Evidence of supressed star formation
relative to the field is present in the whole cluster sample, out to 2 R_{200},
so the mechanism responsible for the differential evolution must be acting at a
large distance from the cluster centre, and not just in the core. The mean star
formation rate in the cluster galaxies with the strongest emission corresponds
to an increase in the total stellar mass of less than about 4% if the star
formation is due to a secondary burst lasting 0.1 Gyr.Comment: aasms4 latex, 3 postscript figures, accepted for publication in ApJ
Letters. Also available at http://astrowww.phys.uvic.ca/~balogh
Transformation of stimulus correlations by the retina
Redundancies and correlations in the responses of sensory neurons seem to
waste neural resources but can carry cues about structured stimuli and may help
the brain to correct for response errors. To assess how the retina negotiates
this tradeoff, we measured simultaneous responses from populations of ganglion
cells presented with natural and artificial stimuli that varied greatly in
correlation structure. We found that pairwise correlations in the retinal
output remained similar across stimuli with widely different spatio-temporal
correlations including white noise and natural movies. Meanwhile, purely
spatial correlations tended to increase correlations in the retinal response.
Responding to more correlated stimuli, ganglion cells had faster temporal
kernels and tended to have stronger surrounds. These properties of individual
cells, along with gain changes that opposed changes in effective contrast at
the ganglion cell input, largely explained the similarity of pairwise
correlations across stimuli where receptive field measurements were possible.Comment: author list corrected in metadat
Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations
The global asymptotic nonlinear behavior of 1 1 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
Nonlinear dynamics and numerical uncertainties in CFD
The application of nonlinear dynamics to improve the understanding of numerical uncertainties in computational fluid dynamics (CFD) is reviewed. Elementary examples in the use of dynamics to explain the nonlinear phenomena and spurious behavior that occur in numerics are given. The role of dynamics in the understanding of long time behavior of numerical integrations and the nonlinear stability, convergence, and reliability of using time-marching, approaches for obtaining steady-state numerical solutions in CFD is explained. The study is complemented with spurious behavior observed in CFD computations
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