Re: simulation +- theory

["John S. Denker" <jsd@MONMOUTH.COM>]

At 11:39 AM 12/29/99 -0500, David Bowman wrote:
> Regarding Ludwik's recently reported results for a rocket problem:
>
> Numerical simulations are not compared to theories in the sense that
> experimental results are compared to theories (i.e. as tests of the
> theories, the experimental results, or some combination of both).

Actually, Ludwik described his situation correctly.  In situations such as
his, e.g. when one is hunting for imperfections in the code, it is very
common and very appropriate to compare the simulation result with the
theoretical result.

> Rather, they are (to the extent the numerical techniques and algorithms
> are reliable solution techniques) predictions *of* those theories.

That's misleading, and gives a naively over-optimistic view of how
simulations are done.  Industrial-strength simulation programs are designed
to have all sorts of theoretical checks, which are often very valuable.

1) To give a specific example:  I once was doing a numerical simulation of
the Schroedinger equation.  The intended purpose was to calculate the ratio
of certain probabilities, and these ratios would have been difficult to
calculate any other way.  Meanwhile the program made many other
predictions, such as the total probability, which according to theory
should have been a conserved constant.  I was able to use the latter as a
sensitive check on the numerical accuracy of the program.

2) There are many other ways in which the agreement of the program with
theory in certain cases can be used to build confidence in the program's
performance in other cases.  It would be unwise to pass up the opportunity
to make such a check.

3) In Ludwik's case, the most important output of the program was not the
numerical prediction.  The important output was the program itself!  The
program constituted a statement of the fundamental physics, stated in a
language more universal and much less ambiguous than English, namely the
computer language.