Stepping methods predictor corrector method

One way to do this is afforded by the predictor-corrector method. We ignore terms higher than those shown explicitly, and calculate the predicted terms starting with bP(t). However, this procedure will not give the correct trajectory because we have not included the force law. This is done at the corrector step. We calculate from the new position rP the force at time t + St and hence the correct acceleration a (t -f 5t). This can be compared with the predicted acceleration aP(f -I- St) to estimate the size of the error in the prediction step... 

An important question is the relative numerical efficiency of the two methods or, more generally, the two families of methods. At a fixed step size the predictor - corrector methods clearly require fewer function evaluations. This does not necessarily means, however, that the predictor - corrector methods are superior in every application. In fact, in our present example increasing the step size leaves the FTunge - Kutta solution almost unchanged, whereas the Milne solution is deteriorating as shown in Table 5.1. 

This system of equations was solved by a predictor-corrector method for several values of a, / , and y using a digital computer. It was not possible to examine values of ft above 50 (a = 0.001, y = 0) as the method broke down because of accumulated errors. Up to these values, although a step is formed in the extent of reaction vs. time curve, the rate of acceleration in the third phase of the reaction was much slower than observed in the experiments. 

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. 

Numerous predictor-corrector methods have been developed over the years. A simple second order predictor-corrector method can be constructed by first approximating the solution at the new time step using the first order explicit Euler method ... 

Predictor-corrector methods [47,48] are appropriate for the integration because they require only one evaluation of the slope for each integration step. Molecular simulation is unusual in the context of the numerical treatment of differential equations, because an approximation to the slope is available before the simulation is complete. This information can be used to update the state point as the simulation proceeds. An increment in a typical GDI series entails the following steps, which for concreteness we describe for an integration in the P-p plane... 

Sixth Order Methods - Four-step Predictor-Corrector Methods. -Simos and Mitsou22 have considered the following family of methods ... 

Simos24 has also considered the following four-step predictor-corrector method ... 

A New Phase Fitted Method. - Consider the one free parameter symmetric three-step hybrid predictor-corrector explicit method ... 

So-called predictor-corrector methods have been used in molecular dynamics from its inception. In this method, molecular forces at time t are used to predict a trial phase point f +i. This prediction requires information about the state of the system at times earlier than f , so that the method is not self-starting. The forces at the phase point f i are then calculated and are used to refine the prediction and generate the actual phase point The method can be formulated to high order (fourth- and fifth-order procedures have been used) and is therefore very accurate. Two evaluations of the forces are required per step. The method is relatively fast given its high accuracy. However, the storage requirements are quite large, so that the method can be inconvenient in some applications. 

At some installations, a particular user might be assigned the use of the computer for a predetermined period of time. During this time, the user will have available all of the resources of the computer system. In such a situation, there is no incentive to minimize the storage requirements, as long as sufficient storage exists to accommodate the job. The most important consideration is the number of time steps that can be calculated within the assigned time period. Therefore, the relative speed of the predictor-corrector method would make it the method of choice under these conditions. 

In practice, implicit multistep methods are used to improve upon approximations obtained by explicit methods. This combination is the so-called predictor-corrector method. Predictor-corrector methods employ a single-step method, such as the Runge-Kutta of order 4, to generate the starting values to an explicit method, such as an Adams-Bashforth. Then the approximation from the explicit method is improved upon by use of an implicit method, such as an Adams-Moulton method. Also, there are variable step size algorithms associated with the predictor-corrector strategy in the literature [5,25]. 

T. E. Simos, A Family of 4-Step Exponentially Fitted Predictor-Corrector Methods for the Numerical-Integration of The Schrodinger-Equation, Journal of Computational and Applied Mathematics, 1995, 58(3), 337-344. 

The choice of the available difference methods is determined by the fact that the force calculation (the evaluation of f) is the most time consuming part of the computation, involving a double summation over particles. This rules out any method of solution (such as Rimge-Kutta and extrapolation methods) that require several fimction evaluations per step. General methods requiring only one force evaluation per step are multi-value predictor-corrector methods. 

The total number of sensitivity equations is equal to the number of state variables times the number of model parameters (i.e. 10x22). These equations are derived in reference 13. The variation of the sensitivity coefficients along the reactor is determined from the numerical integration of the sensitivity differential equations (24) and model equations (1)-(10). It should be noted that the sensitivity equations (24) are extremely stiff. Thus, extra care must be taken in integrating these equations. Accordingly, a multi-step predictor corrector method suitable for stiff differential equations was used. 

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. 

Most numerical path following methods are predictor-corrector methods. The predictor at step i provides a starting value for the Newton iteration which attempts to correct this value to x si) = x. ... 

As mentioned before, the optimum number of application of corrector is two. Therefoi-e, in the case of using a predictor-corrector method, if the convergence is achieved before the second corrected value, the step size may be increased. On the other hand, if the convergence is not achieved after the second application of the corrector, the step size should be reduced. 

There are many variants of the predictor-corrector theme of these, we will only mention the algorithm used by Rahman in the first molecular dynamics simulations with continuous potentials [Rahman 1964]. In this method, the first step is to predict new positions as follows ... 

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