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Topic: Is this an exceptionally hard set of questions to answer?
Replies: 68   Last Post: Nov 11, 2002 7:54 PM

 Messages: [ Previous | Next ]
 Kevin Foltinek Posts: 680 Registered: 12/8/04
Re: Is this an exceptionally hard set of questions to answer?
Posted: Oct 16, 2002 3:51 PM

Alberto Moreira <junkmail@moreira.mv.com> writes:

> >Sometimes numerical solutions "blow up" because of instability in the
> >numerical scheme. (To see this for yourself, discretize the heat
> >equation u_t=u_xx,

> [snip]
> One solution is to use simulation, not computation. This is what I've
> been talking about all the time. The final test for an aircraft
> design, before the maiden flight, is the wind tunnel - equations alone
> won't do. Well, we can set up a simulated wind tunnel inside our
> computers, and we don't necessarily need to put in the Navier-Stokes
> equations, either in their closed form or in their numerical form.

If you're going to simulate a wind tunnel in your computer, then you
are doing computation. Whether the computations you do to simulate
your wind tunnel are based on Navier-Stokes or other models is
irrelevant; the numerical stability and accuracy must be estimated if
you want to use the results.

If you're going to build a prototype and put it into a real wind
tunnel, then you're not doing modelling, you're doing prototyping. If
you want to build a model rather than a prototype, then you are doing
modelling, but you had better understand how things change with the
scale, which requires at least a superficial understanding of the
mathematical model.

> Many blowups due to numerical instability can be negotiated by
> increasing the precision.

You obviously did not investigate the heat equation as I suggested you
do. Increasing the precision *cannot* fix the problems caused by
numerical instability. These problems are much more fundamental than
that. (Blindly subdividing the grid into more squares similarly
cannot fix the problems.)

> >This is not directly relevant to a K-12 education discussion, but it
> >does show that raw computational power may not be sufficient to obtain
> >correct answers from a model.

>
> If we had orders of magnitude more computational power than we have
> today, we could perform the numerical method with an
> infinite-precision numerical package written in Lisp:

And if you don't pay attention to the mathematics of numerical
instability, your solution will blow up in just as many steps as when
you perform the same method with single-precision floats on a 1970s
era computer.

Kevin.

Date Subject Author
9/28/02 Karl M. Bunday
9/30/02 Alberto C Moreira
9/30/02 Shmuel (Seymour J.) Metz
10/5/02 Moufang Loop
10/7/02 Shmuel (Seymour J.) Metz
9/30/02 Stephen Herschkorn
9/30/02 Magi D. Shepley
10/1/02 Karl M. Bunday
10/2/02 Kevin Foltinek
10/2/02 Karl M. Bunday
10/3/02 Alberto C Moreira
10/3/02 Kevin Foltinek
10/3/02 Jim Hunter
10/4/02 Herman Rubin
10/4/02 Alberto C Moreira
10/5/02 Herman Rubin
10/4/02 Alberto C Moreira
10/4/02 Kevin Foltinek
10/5/02 Alberto C Moreira
10/6/02 Virgil
10/6/02 Herman Rubin
10/6/02 Jim Hunter
10/6/02 Virgil
10/7/02 Kevin Foltinek
10/8/02 Alberto C Moreira
10/8/02 Kevin Foltinek
10/9/02 Alberto C Moreira
10/10/02 Kevin Foltinek
10/11/02 Alberto C Moreira
10/14/02 Kevin Foltinek
10/15/02 Alberto C Moreira
10/15/02 Kevin Foltinek
10/16/02 Alberto C Moreira
10/16/02 Kevin Foltinek
10/14/02 Kevin Foltinek
10/16/02 Alberto C Moreira
10/16/02 Kevin Foltinek
10/12/02 Shmuel (Seymour J.) Metz
10/14/02 Kevin Foltinek
10/25/02 Van Bagnol
10/25/02 Alberto C Moreira
10/26/02 Van Bagnol
10/27/02 Alberto C Moreira
10/27/02 Herman Rubin
10/28/02 Kevin Foltinek
10/29/02 Alberto C Moreira
10/24/02 Van Bagnol
10/25/02 Van Bagnol
10/26/02 Alberto C Moreira
10/28/02 Kevin Foltinek
10/29/02 Alberto C Moreira
10/29/02 Kevin Foltinek
10/31/02 Alberto C Moreira
10/31/02 Kevin Foltinek
11/2/02 Alberto C Moreira
11/2/02 David Redmond
11/3/02 Alberto C Moreira
11/3/02 Alberto C Moreira
11/4/02 Kevin Foltinek
11/2/02 Virgil
11/4/02 Kevin Foltinek
11/5/02 Alberto C Moreira
11/5/02 Kevin Foltinek
11/6/02 Alberto C Moreira
11/7/02 Kevin Foltinek
11/9/02 Alberto C Moreira
11/11/02 Kevin Foltinek
10/3/02 Kevin Foltinek
10/5/02 Magi D. Shepley