[Haskell-cafe] Squashing space leaks

[Jerzy Karczmarczuk]

Daniel Fischer wrote:

>If the algorithm - including dt - is prescribed, fine, but I wonder what sort 
>of deviation physicists would consider acceptable.
>For dt = 0.01, k = 2000 we have a relative error of about 2*10^(-5), is that 
>within accepted bounds or not? (Any physicists hang about here?)
>  
>
It is becoming explicitly off-topic...

I didn't follow this thread, I had just one glimpse on the equations
used, I had the impression that I saw the standard Euler extrapolation
instead of something more stable, like, e.g., Verlet, and I switched off,
having other stuff to do.

Now, mind you, *no physicist* will tell you a priori whether this or
that relative error is acceptable or not. 0.01 percent looks nice, but
if you simulate a system in order to see whether it is stable or not,
then it won't suffice. For example: is the Solar system eternal?

On the other hand, if your system is inherently chaotic, ergodic, and
if you are not interested in KAM tori or other islands of stability, then
the system is much more tolerant, one chaos is essentially equivalent
to another chaos, the details of the trajectory are irrelevant. Then, small
errors are not critical at all.

==

What some people do :
1. They compute the initial energy.
2. They solve the differential equations using some *good* methods.
3. After some steps they stop for a moment, they scratch their heads,
   and they recompute the energy.
   If it changed a bit, then they RENORMALIZE the velocity vectors in
   such a way that the energy *remains* constant unconditionally.


Jerzy Karczmarczuk