## References for Chaos Part 0

The literature references cited in Chaos Part 0 are listed in this post. Maybe this will become a Pages.

## Chaos Part 0: ODE Chaotic Response is Numerical Noise

In this post we look at a every important issue relative to the foundations of chaotic response as inferred from numerical calculations with systems of ordinary differential equations (ODEs). The relationship of the contents of these notes to calculations with numerical weather prediction (NWP) and general climate models (GCMs) might be discussed in future posts.

**Summary Statement**

Converged numerical solutions of systems of ordinary differential equations that exhibit chaotic response have never been published. Converged numerical solutions have yet to be obtained for such equation systems. The preceding statements need to be qualified by limiting them to long ranges of values of the independent variable. And of course the statements apply only to the situations for which a given equation system exhibits chaotic response. Additionally, the statements are based on what I will denote as standard numerical solution methods. Standard numerical solution methods include just about all methods that are widely used. Solution methods other than standard, denoted as self-validating, will be specified later in these notes.

Given that chaotic response of complex dynamical systems at its foundation is based almost entirely on numerical solutions of the ODE systems, this situation is almost beyond comprehension. The recently-published paper reviewed here and here on this blog clearly shows the lack of convergence as the step size is refined.

## An Important Peer-Reviewed Paper: Part 1

We’ll now look at some of the results presented in the paper.

**Introduction and Background**

The authors have introduced the subject of convergence of numerical methods into the field of chaotic dynamical systems. This field is very important in many areas of current intense study and investigation. Numerical models and solution methods exhibit chaotic dynamical-system characteristics in weather and climate modeling, direct numerical and large eddy simulations of turbulent flows, as well as the classical studies of chaotic systems through nonlinear ODEs as introduced by Lorenz and others. The author’s paper seems to be the first in the literature to present results of systematic investigations of convergence into this important field of research and applications.

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## An Important Peer-Reviewed Paper: Part 0

This paper Time-step Sensitivity of Nonlinear Atmospheric Models: Numerical Convergence, Truncation Error Growth and Ensemble Design addresses some of the important issues for which this blog was established. Extensive discussions will follow as I get the results of my work documented. I have used the 3D Lorenz equations for most of my work, but have looked at other ODEs for which chaotic response has been demonstrated.

One conclusion that I am almost certain about at this time is that standard numerical solution methods applied to ODEs that exhibit chaotic responses have never been shown to converge. Additionally, it is very likely that convergence can not be demonstrated. The calculated numbers are very likely noise that does not satisfy the continuous equations. Some parts of this conclusion will cary over to numerical solution methods for PDEs. Some of the issues were mentioned in this post. The chaotic responses calculated by AOLGCM models/codes are in fact purely numerical artifacts from a combination of the numerical solution methods, lack of convergence of the calculations, and the algebraic parameterizations used in the models.

These issues will be addressed in subsequent posts here. First we take a preliminary look at some of the issues brought to light by the subject paper.

## A Start on a V&V and SQA Bibliography

Here is a start on a bibliography for V&V and SQA books and articles.

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## Lack of Convergence, Under-Resolution, and Numerical Errors

**The Basic Hypotheses**

The following is well known but because it is the focus of this discussion I list it for handy reference.

**Convergence is Paramount; Nothing else Counts**

The fundamental objective of numerical solution of systems of algebraic equations, ordinary differential equations (ODEs) and partial differential equations (PDEs) is to ensure that the approximations made in order to solve the equations do not in fact influence the solutions. In the case of systems of algebraic equations, it must be shown that the stopping criteria applied to iterative solution methods does not influence the accepted solutions. The solutions are independent, to an acceptable level, of the stopping criteria, and the calculated numbers satisfy the equations.

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