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.
Excellent First Step
The authors of the paper have presented an excellent first step into investigations of the status of the accuracy of standard numerical solution methods when applied to simple systems of nonlinear ODEs that exhibit chaotic responses. The additional information regarding numerical solutions of the systems of algebraic equations arising from discrete approximations to PDEs is also welcomed.
Consistency, stability, and convergence for discrete approximations to continuous equations are of fundamental importance in all studies of mathematical models of physical phenomena and processes. And, while of undeniable significance, these issues are frequently overlooked, or completely neglected, in the rush to get a computer model into the hands of users. This is a very unfortunate situation and generally leads to difficulties down the road after a code is put out for production applications. Consistency, stability, and convergence of numerical approximations to continuous model equations was the subject of a previous post on this blog. The overriding importance of convergence was discussed there. The present paper has made clear the importance of convergence.
The paper is an excellent example of some of the activities that are required to be determined prior to coding the mathematical models and solution methods into a production-level computer program. Additional discussions of these activities are given below.
In this the first of a couple of posts on several aspects of numerical solution methods, we will expand on some of the issues addressed in the paper. In later posts, additional new information regarding convergence of numerical solutions of nonlinear equation systems that exhibit chaotic responses will be covered. A short bibliography of papers and reports in which the importance of careful attention to numerical solution methods has been given here.
A More Nearly Complete Program
A more nearly complete program that is required prior to coding at the production-code level is outlined in the following paragraphs. The activities discussed in the paper under review fulfill one of these activities as indicated below. A more nearly complete pre-coding program includes the following procedures:
(a) Derive the final versions of the continuous equations. Let’s take the equations for fluid motions as an example and call them the Navier-Stokes equations. The final model equations will very likely be simplifications of the general content of the fundamental basic equations. While these latter equations are statements of conservation of mass, momentum and energy, the model equations are just that; models of conservation of mass, momentum, and energy. The model equations cannot conserve mass, momentum, and energy because the applied assumptions and approximations have destroyed that property from the model equation system.
The model equations represent hypotheses that will later be determined to be correct or not by Validation of calculations with the equations against comparisons with empirical data. Validation and Qualification of the model/code/user for applications for which it was designed follows these preliminary activities far down the road.
(b) After the model equations are determined/derived, the characteristics, as in characteristics of PDEs are determined. The characteristics are used to determine the classification for the continuous equations (parabolic, hyperbolic, elliptic), and the correct number and kind of boundary conditions that can be used given the solution domain for the problem, so that a well-posed problem in the continuous domain is set.
( c) With the model equations and boundary conditions, the next steps are associated with development of numerical solution methods. There’s tons o’ art involved here and the decision can be dictated by considerations about resolution of temporal and spatial scales of importance in the applications of the model. Computer run time and CPU-power requirements might also be considerations. Whatever the important and significant decision-drivers, numerical solution methods are determined.
(d) The next required, important, and significant steps are associated with determinations of the properties and characteristics of the solution methods for the discrete form of the model equations and proper and correct boundary conditions. The properties and characteristics include (i) consistency, (ii) stability, and (iii) convergence. The discrete approximations of the continuous equations are firstly required to be shown to be consistent with the continuous equations and associated boundary conditions.
(e) Following determination of consistency, the proposed solution method(s) for the discrete equations are required to be shown to be stable. The properties of consistency and stability are generally accepted to imply convergence; especially for well-posed linear model problems, and also especially for well-posed non-linear model problems that exhibit physical instability. The nature of the instabilities must be shown to be associated with the physics captured in the continuous equations and shown not to be a property of the numerical solution methods. There are many excellent textbooks in which the required properties of numerical solution methods are discussed in detail.
(f) Following coding of the numerical solution method, most likely in a test-bed toy-model code, the required characteristics of consistency, stability and convergence can be checked by use of calculations, and maybe the use of the Method of Manufactured Solutions (MMS). Stability and convergence can be checked by use of calculations, for examples.
Importantly, testing and demonstration of the asymptotic order of the rate of convergence of the numerical methods can be performed. As in the case of the subject paper, there is a potential for surprises to be discovered in the results of this activity. Surprises have also been the results when testing of stability is conducted. And for convergence, too, as again shown in the subject paper.
Successful completion of task (a) through (f) generally results in a system of discrete approximations to the continuous equations and associated numerical solution methods that are ready for coding into production-level software. This latter task brings a whole other very wide range of additional issues that are way beyond the scope of discussions here.
First Conclusions
The subject paper presents a focus of Step (e) above. Unfortunately, the procedure has been applied long after the production-level codes have been designed, developed, and released. The suggestion that the temporal step size be used as an additional parameter, a suggestion that cannot be allowed to stand, is one of the surprise consequences of not preceding production-level codes with analysis and testing.
In subsequent posts we will discuss the significance of other results presented in the paper.
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