8.4 Integration methods: symplectic integrators 215The fact that this quantity is conserved can also be checked directly using(8.34b).
This energy is equal to 1/ 2 −h^2 /8 for the solution cos(ωt)withωgiven inEq. (8.33).
For general potentials, the discrete energy is not known.
As mentioned before, the absence of drift in the energy in the case of the harmonic
oscillator can be explained by the time-reversibility of the Verlet algorithm, and
comparisons with Runge–Kutta integrators for example, which are in general not
time-reversible for potentials such as the harmonic oscillator, do not convincingly
demonstrate the necessity for using a symplectic algorithm. Symplecticity does
however impose a restriction on the noise, but time-reversibility does not.
Symplectic integrators are generally recommended for integrating dynamical
systems because they generate solutions with the same geometric properties in
phase space as the solutions of the continuum dynamical system. The fact that the
deviation of the energy is always bounded is a pleasant property of symplectic
integrators. Symplectic integrators are considered in more detail inSection 8.4.2.
Finite precision of computer arithmetic obviously does not respect the symplectic
geometry in phase space. Hockney and Eastwood observed that when numbers are
rounded off properly in the computer, the system tends to heat up because the
rounding effects can be viewed as small random forces acting on the particles[19].
If real numbers are systematically truncated to finite precision numbers, the system
cools down slowly. Both effects are clearly signs of nonsymplectic behaviour.
Several classes of symplectic integrators with explicit formulas for different
orders of accuracy have been found. Runge–Kutta–Nystrom integrators (not to be
confused with ordinary Runge–Kutta algorithms) have been studied by Okunbor
and Skeel[ 20 ].Yoshida[21]and Forest[22]have considered Lie-integrators. Their
approach follows rather naturally from the structure of the symplectic group, as we
shall see inSection 8.4.2.^3
Let us make an inventory of relevant symmetry properties of integrators. First of
all, time-reversibility is important. If it is present in the equations of motion, as is
usually the case in MD, it is natural to require it in the integration method. Another
symmetry is phase space conservation. This is a property of the trajectories of the
continuum equations of motion – this property is given by Liouville’s theorem –
and it is useful to have our numerical trajectories obeying this condition too (note
that time-reversibility by itself does not guarantee phase space conservation). The
most detailed symmetry requirement is symplecticity, which will be considered in
greater detail below (Section 8.4.2). This incorporates phase space conservation and
conservation of a number of conserved quantities, the so-calledPoincaré invariants.
The symplectic symmetry properties can also be formulated in geometrical terms
(^3) Gear algorithms [16, 23, 24] have been fashionable for MD simulations. These are predictor–corrector
algorithms requiring only one force evaluation per time step. Gear algorithms are not symplectic and they are
becoming less popular for that reason.