Re: CCL:Multi-time scale integration.

[Richard Gillilan <richard |-at-| tc.cornell.edu>]

 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 few outdated refs on this topic, but I seem to recall some passing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