Re: [Phys-l] accurate numerical solution of equations of motion

[John Denker <jsd@av8n.com>]

On 10/22/09 06:39, John Mallinckrodt wrote:

> On the other hand, I had not used Denker's prescription of the Euler  
> method, but rather the so-called Euler-Cromer method [see "Stable  
> solutions using the Euler approximation," Alan Cromer, 49 (5), 455-9  
> (1981)] in which Denker's Eq 2 is replaced with
>
> x(3) = x(2) + v(3) Δt
>
> This method is so vastly superior to the standard Euler method and so  
> simple to implement that I tend not even to remember that I'm doing  
> it any more. 


1) Verrry interesting.  Thanks.

2) In addition to Cromer's AJP article mentioned above, i.e.
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000049000005000455000001&idtype=cvips&gifs=yes

see also
  http://en.wikipedia.org/wiki/Semi-implicit_Euler_method

and especially
  http://en.wikipedia.org/wiki/Symplectic_integrator



In simplest terms, the Euler-Cromer method can be described as 
 1) using ordinary forward Euler to find v by integrating a, then
 2) using _backward_ Euler to find x by integrating v.

There is no need for second-order Euler or for stride=2.

I don't claim to understand this yet, but I'm certainly not 
going to rest until I do.  Evidently the errors introduced
in step 1 are canceled to leading order by the errors
introduced in step 2.  More importantly, the two steps
together conserve area in phase space.  That is to say, 
Liouville's theorem is upheld.  (Is it upheld _exactly_?
I can't say, yet.)  This is super-important; remember that 
Liouville's theorem is the essence of the second law of 
thermodynamics, and the Heisenberg uncertainty principle, 
and the brightness theorem.  The wikipedia article mentions 
the two-form dp /\ dq but forgets to mention that it represents 
area in phase space.

Previously I recommended testing such methods on the Kepler
problem.  You can also test it on the harmonic oscillator,
which is even simpler (one dimension instead of two).  If
you're going to plot something, plot the phase space of
the oscillator (p versus x).  See e.g.
  http://www.physics.udel.edu/~jim/Ordinary%20Differential%20Equations/Euler-Cromer%20Method.htm
although that won't tell you much beyond a subset of what 
the previously-mentioned references said.


On 10/22/09 09:06, Rick Tarara wrote:
 
> > Next step is to use their spreadsheet to predict the behavior of a falling 
> > foam ball.  This they drop and time for distance up to 10-12 meters (drop 
> > off balconies in our library).  THEN display the data and the predictions on 
> > the same graph--again they haven't done this before--and search on the 
> > adjustable parameters for the best fit.
> >
> > I try to emphasize, especially to the engineers, that this is probably the 
> > most useful and important lab exercise we will do all year!

Sounds good to me!