Predictor-corrector algorithm

  Predictor-corrector algorithms constitute another commonly used class of methods to integrate the equations of motion. Those more often used in molecular dynamics (see e.g. [12]) are due to Gear, and consists of three steps:

1.

Predictor. From the positions and their time derivatives up to a certain order q, all known at time t, one ``predicts'' the same quantities at time $t+\Delta t$ by means of a Taylor expansion. Among these quantities are, of course, accelerations $\bf a$.

2.

Force evaluation. The force is computed taking the gradient of the potential at the predicted positions. The resulting acceleration will be in general different from the ``predicted acceleration''. The difference between the two constitutes an ``error signal''.

3.

Corrector. This error signal is used to ``correct'' positions and their derivatives. All the corrections are proportional to the error signal, the coefficient of proportionality being a ``magic number'' determined to maximize the stability of the algorithm.

For more details the reader is referred to [3], §3.2, or [6], §4.4. A detailed comparison between the Verlet and the Gear scheme can be found in [19].