Next: Molecular dynamics at constant Up: Equations of motion Previous: The leapfrog algorithm

## The Predictor-Corrector algorithm

We would like to solve the second-order differential equation

 (8.8)

where is related to the forces in equation (8.2) by . The first step of this algorithm consists in evaluating the atomic positions and velocities at time from the positions and the velocities at time , where i = 0,...,k-2, k being the order of the predictor part. The extrapolation is given by

 (8.9)

for the atomic positions, and

 (8.10)

for the velocities. The coefficients and satisfy the equation
 (8.11)

These predicted values are then corrected from the value of at a time (calculated from the predicted values themselves), using the expressions

 (8.12)

for the atomic positions, and

 (8.13)

for the velocities. The coefficient and satisfy similar equations as and . The coefficient used in the present work are for k = 4, and are given in table 8.1.

Table 8.1: Coefficient of the Predictor-Corrector algorithm for k = 4 for second-order differential equation.
 k = 4 ( 1/24) 1 2 3 19 -10 3 27 -22 7 3 10 -1 7 6 -1

The Predictor-Corrector algorithm gives more accurate positions and velocities than the leapfrog algorithm, and is therefore suitable in very delicate'' calculations. However, it is computationally expensive and needs significant storage (the forces at the last two stages, and the coordinated and velocities at the last step).

Next: Molecular dynamics at constant Up: Equations of motion Previous: The leapfrog algorithm