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L. S. Yao and D. Hughes, “Comment on ‘Time Step Sensitivity of Nonlinear Atmospheric Models: Numerical Convergence, Truncation Error Growth, and Ensemble Design’,” Journal of the Atmospheric Sciences, Vol. 65, No. 2, February 2007, pp. 681-682.

has been cited by the following article:

  • TITLE: Convergence Criterium of Numerical Chaotic Solutions Based on Statistical Measures

    AUTHORS: Julio Cesar Bastos de Figueiredo, Luis Diambra, Coraci Pereira Malta

    KEYWORDS: Chaotic Attractor, Statistical Measure, Numerical Integration

    JOURNAL NAME: Applied Mathematics, Vol.2 No.4, March 31, 2011

    ABSTRACT: Solutions of most nonlinear differential equations have to be obtained numerically. The time series obtained by numerical integration will be a solution of the differential equation only if it is independent of the integration step h. A numerical solution is considered to have converged, when the difference between the time series for steps h and h/2 becomes smaller as h decreases. Unfortunately, this convergence criterium is unsuitable in the case of a chaotic solution, due to the extreme sensitivity to initial conditions that is characteristic of this kind of solution. We present here a criterium of convergence that involves the comparison of the attractors associated to the time series for integration time steps h and h/2. We show that the probability that the chaotic attractors associated to these time series are the same increases monotonically as the integration step h is decreased. The comparison of attractors is made using 1) the method of correlation integral, and 2) the method of statistical distance of probability distributions.