Gear predictor corrector method

The integral time interval is about 1 fs and the equation of motion is integrated by fifth-order Gear predictor-corrector method [8]. 

Rigorous and stiff batch distillation models considering mass and energy balances, column holdup and physical properties result in a coupled system of DAEs. Solution of such model equations without any reformulation was developed by Gear (1971) and Hindmarsh (1980) based on Backward Differentiation Formula (BDF). BDF methods are basically predictor-corrector methods. At each step a prediction is made of the differential variable at the next point in time. A correction procedure corrects the prediction. If the difference between the predicted and corrected states is less than the required local error, the step is accepted. Otherwise the step length is reduced and another attempt is made. The step length may also be increased if possible and the order of prediction is changed when this seems useful. 

The differential equations which arise in almost all chemical kinetic studies of complex reaction schemes are "stiff differential equations". In chemical kinetics the stiffness is caused by the huge differences in the reaction rate constants of the various elementary reactions. It is impossible to solve such a system of differential equations by the usual Rung-Kutta methods. Therefore we used a program, described by Gear (J ) as a special multi step predictor-corrector method with self adjusting optimum step size control. 

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