More on ODEs, MMS and Ill-Posed IVPs
First for the nomenclature: ODEs means Ordinary Differential Equations, MMS means the Method of Manufactured Solutions, and IVPs means Initial Value Problems.
This previous post provided some information on these subjects. So far as I know, that post presented the first results for application of MMS to ill-posed IVPs. That post suggested that for the numerical solution methods used therein, the original Lorenz system of 1963 has yet to be correctly solved.
I have some additional results, a summary of which is:
I think the Lorenz system has not yet been accurately integrated by any numerical solution methods. Higher-order methods plus, at the same time, higher precision representation of numbers will give results that might appear to be solutions. But, calculations for sufficiently long time spans will show that errors always increase.
I’ve uploaded a file here.
ILL-Posed IVPs and the MMS
The subject means ill-posed Initial Value Problems (IVPs) and the Method of Manufactured Solutions (MMS). I have uploaded two files; this one has the words, and this one has the figures. If you open them in separate windows the text is easier to follow.
All comments, especially corrections for incorrectos, will be appreciated.
Energy and the Lorenz System
Introduction
I’ve decided to modify this post and put an example here. Examples have the potential to provide more understanding of the important technical issues.
So, let’s say it’s Saturday January 5, 2008, at 4:30 am and a Butterfly is sitting on the railing of the deck outside the house. Actually, the railing is snow-covered and the Butterfly is sitting on the snow. The air is still, the sky is crystal-clear, there is significant radiative cooling underway and the temperature is dropping like a rock; it’s well below zero in both C and F. The Butterfly uses one wing to stifle a yawn and that wing moves slowly toward its mouth and then back to its resting place; the Butterfly needs the cover for warmth.
Here’s the question. What effect will that flap of the Butterfly’s wing have on the potential for a hurricane to form in the Gulf of Mexico in July 2008.
Some of the technical issues behind this question are the subjects of this post and possibly one or two others in future.
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.
Models Methods Software 