From a physics blog thread on public involvement in science:

this lecture from a geologist :

> "Paul, as we’ve discussed before, there are real-world situations which can’t be reduced to a closed-form solution of an accepted physics model. In most of what we’re talking about, the problem is not the use or non-use of accepted physics, but that properly representing the real world using accepted physics results in situations where there is no closed-form solution. Sometimes Nature just doesn’t cooperate. I’ve seen such situations in my professional life, and others have been discussed on this blog, where strict adherence to models with closed-form solutions restricts you to a world that is so different from the real world it might as well be an undiscovered exoplanet. For example, economic models that don’t properly represent the carbon cycle and reach optimistic conclusions about mitigation requirements because their model draws down atmospheric CO2 an order of magnitude faster than the real world can.

> In those situations you have three choices:

> 1) Model the fantasy world using closed-form solutions and congratulate yourself you’ve avoided any numerical pitfalls. But not be surprised when people say “meh, it’s just a fantasy world, look how drastically it diverges from reality if we omit the most recent ten years of data and hindcast”.

> 2) Model the real world numerically, while being cognisant of the potential pitfalls and taking steps to mitigate them. Independent reproduction is better for that than Auditing, for the pitfalls as well as the science. Auditing will just find the pitfalls that reside in library solvers or hardware differences. Different code, different solver, same or similar data may identify structural problems.

> 3) Give up."


What's interesting about this argument is how backward the logic is. A closed-form analytical solution can significantly help in checking the validity of a numerical solution to a given set of equations. One of the current issues of Navier-Stokes numerical solutions is in how sensitive they are to grid and time-stepping resolution. The closed-form solution can provide a guide to how close the numerical solution is converging to the set of asymptotic values.

I can see how this would evolve given a solution such as cos(*A* sin(*w t*)). If *A* gets large enough, the numerical solution requires the equivalent of a long Taylor's series of coefficients to capture the full resolution in time that the folding dynamics will reveal. One will know what time-stepping delta is required quite quickly.