From chemistry-request@www.ccl.net Tue Sep 15 09:34:34 1998 Received: from theory.tc.cornell.edu (THEORY.TC.CORNELL.EDU [128.84.30.174]) by www.ccl.net (8.8.3/8.8.6/OSC/CCL 1.0) with ESMTP id JAA18561 Tue, 15 Sep 1998 09:34:34 -0400 (EDT) Received: from ren.tc.cornell.edu (REN.TC.CORNELL.EDU [128.84.244.22]) by theory.tc.cornell.edu (8.8.4/8.8.3/CTC-1.0) with SMTP id JAA20333 for ; Tue, 15 Sep 1998 09:34:35 -0400 Sender: richard@tc.cornell.edu Message-ID: <35FE6CEB.794B@tc.cornell.edu> Date: Tue, 15 Sep 1998 09:34:35 -0400 From: Richard Gillilan X-Mailer: Mozilla 2.02 (X11; I; IRIX 5.3 IP22) MIME-Version: 1.0 To: chemistry@www.ccl.net Subject: Re: CCL:Multi-time scale integration. References: <9809131414.ZM3019@hamilton.math.missouri.edu> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit 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 Sorry, I can't seem to locate my few outdated refs on this topic, but I seem to recall some passing comments in a recent Klaus Schulten paper.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@tc.cornell.edu