Models, Methods, Software

I have started working on a toy model and plan to include analytical sensitivity analysis as part of the methods.  These notes, and an associated extended discussion that I have up-loaded, serve as a short introduction to the subject.

The file is here:

Summary

These notes introduce a few of the ideas and concepts associated with sensitivity analysis for algebraic and ordinary differential equations. By sensitivity I mean what are the effects of changes in the numerical values of the parameters in a system of equations relative to a response function of interest. The response function can take any mathematical form, but I will focus on the values of the dependent variables of the equation system.

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of equation systems that do not possess chaotic response.

Executive Summary
The original PDEs that describe the Rayleigh-Benard convection problem do not posses chaotic behavior. The chaotic response observed with Lorenz-like low-order models (LOM) obtained via mode expansions disappears whenever sufficient resolution is used in the numerical solution methods applied to the original PDEs.

The low order model of the Lorenz equations omits the terms that are responsible for interaction between smaller scales and the large scales. The very interactions that form the basis for invoking the turbulence analogy.

GCMs are consistent with the chaotic response obtained from incorrect low-order models (LOM) expansions of PDEs.

GCMs are consistent with the chaotic response obtained from incorrect solutions to ODEs and PDEs.

GCMs are consistent with the chaotic response observed whenever insufficient resolution is used with numerical solution methods.

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Introduction
The calculations preformed with the equation systems are summarized in the following discussions. The focus had been on testing for convergence of the numerical solution methods to solutions of the continuous equations. By convergence I mean that as the size of the discrete increment for the independent variable is reduced the calculated values for all dependent variables approach limiting constant values for all values of the independent variable.

None of the systems that are said to exhibit chaotic response have shown convergence. One of those, the Terman system, exhibits periodic response, not chaotic response. The Saltzman system was never intended to be an example for chaotic response.

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The numerical solution methods that will be used to check convergence are given in a file that I uploaded.

Let me know if you see any typos or if you want to see some results for a specific equation system.

I’m thinking that Part 1d will be some numerical results.

The equation systems that will be used to check convergence are given in a file that I uploaded. I had tons o’ links and cross references and other good stuff but nothing worked out. Maybe later.

Let me know if you see any typos or if you want to see some results for a specific equation system.

I’m thinking that Part 1c will be the numerical methods.

UPDATE Nov 19, 2007: I have replaced the original uploaded file with a version that has some identification for me in it.

I have way too much material for a single post. I have spent days trying to force a good fit for all the material into a single document. I have put that aside for a while. So these discussions will be broken into several parts. At some future time I might try to tie all the pieces together by use of HTML/PDF.

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The NWP and GCM communities cannot think that a Butterfly will have any influence whatsoever on any physical phenomena or processes of interest. Instead the phenomenology of The Butterfly Effect as exhibited by the numerical calculations of some systems of ordinary differential equations is invoked by hypothesis into NWP and GCM models/methods/codes. I think we need to limit discussions to the Lorenz-like systems of ODEs, as these seem to be the basis for invoking the phenomenology into the NWP and GCM communities. Otherwise we will get side-tracked into discussions of the “chaotic response of complex dynamical systems” in general.

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

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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.

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The chaotic phenomenology of small systems of non-linear ODEs is entirely numerical ODE chaos. And, the original Lorenz system of 1963 contains no physical phenomena or processes of interest in NWP and AOLGCM applications.

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