Subject: Re: CCL:beyond Verlet's algorithm

From: Michael Brunsteiner <[email-masked]>
Date: Tue, 07 Aug 2001 13:45:06 +0100

Thomas Heine wrote:
> 
> Dear CCLers,
> 
> many MD codes use Verlet's algorithm to obtain trajectories in MD
> simulations. This algorithm is very compact and optimized for
> - minimal storage of information
> - minimal number of operations in the integration process
> to advance coordinates (and velocities) with time.
> 
> Let's assume that the computing time needed to get new coordinates R(t+ delta
> t) is negligible to the time spent for calculating energy and forces.
> 
> My question is whether there is a more sophisticated algorithm around, which
> integrates with an as large as possible delta t (determined with some
> theshold), which uses coordinates, velocities and accelerations of N previous
> iterations?

There's, e.g., different kinds of "predictor corrector" and 
"Runge-Kutta" algorithms (see "Numerical Recipies", W. H. Press)

however, low memory consumption and fast execution
are not the only advantages of the Verlet (or the equivalent
velocity Verlet and leap frog) algorithm.
It is also a so called symplectic/time-reversible algorithm
with the important consequence that the total energy is
well conserverd in the course of an MD simulation.
When performing longish MD runs with higher order algorithms 
you usually have to use a quite small time step to avoid
heating up of the the system (unless of course you use a 
thermostate, but it might be physically rather questionable to
use a thermostate for such a purpose).

I believe that, unless you have really good reasons for using them,
the effort involved in implementing such higher order algorithms
wouldn't pay out.

regards
mic

===================================================================
What I like most about myself is that I'm so understanding
when I mess things up.

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Michael Brunsteiner
Centre for Theoretical and Computational Chemistry
University College London
mailto:[email-masked]
http://www.ucl.ac.uk/~uccambr/
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From: Konrad Hinsen <[email-masked]>
Date: Tue, 7 Aug 2001 14:45:09 +0200

> My question is whether there is a more sophisticated algorithm
> around, which integrates with an as large as possible delta t
> (determined with some theshold), which uses coordinates, velocities
> and accelerations of N previous iterations?

Various integrators known from numerical analysis have been tried, but
although some of them have a better accuracy on short time scales,
they all tend to be less stable on long time scales. A lot of
literature exists on that topic, the keyword to look for is
"symplectic integrators". There are other symplectic integrators than
the Verlet algorithm, and some of them permit larger time steps (e.g.
multiple time step integrators), but not in the way you describe, and
not without "help" from the energy evaluation.

Note also that MD integration is more of a black art than science.
Most simulations are performed with time steps that are much too large
compared to the frequency of the system, using standard criteria from
the theory of differential equations. And indeed such time steps do
not give good results for e.g. single molecules, a large system is
required to make them acceptable, probably due to error cancellation.
Moreover, it is well known (but rarely discussed) that of all the
theoretically conserved quantities, only the total energy is preserved
reasonably well in MD. Momentum and (for non-periodic systems) angular
momentum are usually "reset" periodically to prevent the accumulation
of deviations. A much smaller time step will give good conservation of
all these quantities, but is not used because of excessive CPU times.
-- 
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