Multiple time scale integration: An illustrative example

Many physical systems are characterized by motion on many time scales. Often, at least some of these arise from the fundamental nature of the specified interactions. In molecular systems, for example, there will be a fast time scale arising from the internal motion within the molecule (bond and bend vibrations, dihedral angle motion, etc.), an intermediate range time scale arising from the short-ranged strongly repulsive intermolecular interactions, and a slow time scale arising from long-range interactions. These time scales must be buried in the forces, and, therefore, it is clear that the time step in a numerical integration algorithm must be limited by the fastest time scale present. This means that many time steps must be taken before the slower parts of the force change considerably, and this can lead to a large computational overhead. On the other hand, suppose we could exploit the basic physics of the system to devise a numerical integration scheme that treats each part of the force on its own intrinsic time scale. Then, slow forces could be computed less frequently than fast forces. This would be a great advantage, since it is often the case that the fast forces are simple and cheap to evaluate, while the slower forces are the computationally intensive ones.

As an illustrative example, consider the problem of integrating the equations of motion for a system of oxygen molecules in the liquid phase. The figure illustrates a single O molecule held together by a specified internal interaction:

 

Of course, in reality, it is the distribution of electrons in the bond that determines the bonding interaction, however, we are interested in constructing a simple model. One possibility is to introduce a simple harmonic interaction of the form

where is the eqilibrium bond length. The total intramolecular potential is then a sum of harmonic terms over all the molecules:

where is the bond length of each molecule. In addition to the intramolecule component, an intermolecular component is needed to specify how two molecules interact, as illustrated below:

 

The intermolecule interaction occurs between atoms connected by blue lines, i.e. four interactions. Each interaction depends only on the distance between the atoms, and a typical form of the interaction is a simple Lennard-Jones potential:

The total intermolecule potential is then taken as a sum over all pairs of atoms excluding those that are in the same molecule:

where the prime on the summation indicates that some pairs are excluded.

Now, the total potential, U is a sum of the intramolecular and intermolecular contributions:

and the force on each atom is obtained by differentiating U with respect to the position of an atom:

From the above, we see that the force will be composed of two contributions,

Moreover, intramolecular interactions are typically of much higher frequency than intermolecular interactions, which means that the intramolecular motion is fast and only weakly coupled to the intermolecular motion. Finally, if there are N atoms in the system, the number of intramolecular interactions is N/2 while the number of intermolecular interactions is (at most) N(N-2)/2. Thus, the intramolecular interactions will be much cheaper to evaluate than the intermolecular interactions.

The basic idea of multiple time scale integration is to exploit this intrinsic separation of time scales to devise an integration algorithm that treats each time scale with its own characteristic time step.