Models, Methods, Software

When the flows mentioned in the title are approached by use of model conservation/balance equations for the individual constituents, or separate fluid regions, in the flow field, many (all?) are ill-posed as hyperbolic mathematical problems. We can get into issues associated with parabolic and elliptic systems, but let’s try to stick to the hyperbolic case to start off. In fact, one method to regularize (?) the hyperbolic case is to convert it to the parabolic case by use of a vanishingly small viscosity-like parameter. And by all means let’s avoid discussions of the effects of what discrete approximations to the continuous equations do to the whole messy situation.

Why is that?

Could it be that the fundamental continuum mechanics equations for fluid flow, usually taken to be the Navier-Stokes equations, are missing something when physical interfaces are present in the flow field? To be specific, the Euler equations, I think, are the hyperbolic case of interest.

Let’s attempt to count the number of characteristics that are pointing to a physical interface between two constituents or regions. We can do this even when the constituents are red and blue water, for example. Then we’ll have to find enough equations for the unknowns we count. That’ll be the fun part.

Who wants to start?

Let me know if you find incorrectos in the above.

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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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.

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