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Date: Tue, 15 Sep 1998 09:34:35 -0400
From: Richard Gillilan <richard #at# tc.cornell.edu>
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Subject: Re: CCL:Multi-time scale integration.
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Don Steiger wrote:
> 
> In the literature on multi-time scale integration, a lot of importance is
> placed
> on time reversibility.  Why this is so important is something that I have never
> been able to figure out.  If an integration algorithm is efficient and produces
> a small global error then why is this not sufficient.  If anybody can enlighten
> me on this subject I would appreciate it.
> 

My understanding is that some integrators have the additional 
property of being exact symplectic transformations of phase
space from one time step to the next. Technically this means that 
phase-space volume and a hierarchy of other differential
forms discovered by Poincare are exactly preserved under this 
transformation. If I'm not mistaken, this property also implies
exact time reversibility. It has long been thought that the extraordinary 
long-time stability of Verlet's leapfrog algorithm was due to this
property although I have not kept up with the literature enough to 
know if the connection has a formal basis. I believe there have been
a number of numerical experiments in the literature in which high-order
non-symplectic integrators are shown to blow up long before Verlet.
Higher-order symplectic integrators seem to be implicit and so not
as efficient unfortunately. There is a growing literature on this 
subject in Physics I think. Sorry, I can't seem to locate my
few outdated refs on this topic, but I seem to recall some passing 
comments in a recent Klaus Schulten paper. 


Richard Gillilan
Cornell Theory Center
richard /at\tc.cornell.edu