Gear predictor-corrector algorithm

The above formulation produces a set of differential and algebraic equations. These equations are solved by utilizing a Gear predictor-corrector algorithm [16-19]. The predictor operations estimate the system response at the ne rt time step (n+1) based on the response at previous points. A polynomial of order k is fitted to the previous values of the system states of each of the generalized coordinates of the system, and the polynomial thus calculated is used to evaluate a truncated Taylor series expansion to obtain the system response at time n+1... 

The other main class of algorithms are the predictor-corrector algorithms (Gear, 1971). New positions, velocities, accelerations and higher time derivatives of r at (n + l)At are predicted using Taylor expansions and the current values at nAt. But these are not correct, and will eventually fail, because the forces have not been updated. So the accelerations at (n + l)Af are now calculated using the predicted positions, and hence the forces at (n + l)Af, and... 

These correction terms involve numerical coefficients (Gear, 1971 see also Allen and Tildesley, 1987) chosen to give optimum stability and accuracy. Ideally, the corrector step would be repeated to improve the accuracy of the estimates at (n+ l)Af but each correction involves a new evaluation of the forces, which is the most time-consuming part of an MD simulation. So in practice just one or two corrector steps are carried out. Other forms of predictor-corrector algorithm exist. A discussion of various MD algorithms has been given by Berendsen and van Gunsteren (1986). 

In predictor-corrector algorithms time derivatives of the position vectors at time t are used to predict the positions and their derivatives at time H- At. The predicted variables then are corrected according to the difference from those at time t, where a set of Gear constants are used. The latter are chosen to balance accuracy and stability, that is, short- and long-time conservation of energy. Optimized values depend on the order of the Taylor expansion ( order of the algorithm ). 

In 1971, Bill Gear developed what is now called the four-value predictor-corrector algorithm. There are two steps for propagating the positions, velocities, accelerations, and third derivatives of positions. First is the prediction step, which is implemented as follows ... 

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