Predictor corrector method

1 Implicit Fourth Order Methods. - Simos17 has considered the following method  

Using (32) an exponentially-fitted method has been produced (see reference 17, which integrates exactly  

Thomas, Mitsou and Simos18 have considered the following family of methods  

Using (34), another group of exponentially-fitted methods has been produced (see ref. 18). These integrate exactly the following functions  

Sunos19 has derived the final exponentially-fitted method in this group which integrates exactly  

If we truncate up to the fourth term in the RHS of Eq. 7.105, we obtain the following fourth order Adams-Moulton method. 

The common factor in the implicit Euler, the trapezoidal (Crank-Nicolson), and the Adams-Moulton methods is simply their recursive nature, which are nonlinear algebraic equations with respect to y +j and hence must be solved numerically this is done in practice by using some variant of the Newton-Raphson method or the successive substitution technique (Appendix A). 

8 PREDICTOR-CORRECTOR METHODS AND RUNGE-KUTTA METHODS 

A compromise between the explicit and implicit methods is the predictor-corrector technique, where the explicit method is used to obtain a first estimate of and this estimated y +i is then used in the RHS of the implicit formula. The result is the corrected y + i, which should be a better estimate to the true y +i than the first estimate. The corrector formula may be applied several times (i.e., successive substitution) until the convergence criterion (7.49) is achieved. Generally, predictor-corrector pairs are chosen such that they have truncation errors of approximately the same degree in h but with a difference in sign, so that the truncation errors compensate one another. 

One of the most popular predictor-corrector methods is the fourth order Adams-Bashford and Adams-Moulton formula. 

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All these observations tend to favour the Verlet algoritlnn in one fonn or another, and we look closely at this in the following sections. For historical reasons only, we mention the more general class of predictor-corrector methods which have been optimized for classical mechanics simulations, [40, 4T] further details are available elsewhere [7, 42, 43]. 

Thus we find that the choice of quaternion variables introduces barriers to efficient symplectic-reversible discretization, typically forcing us to use some off-the-shelf explicit numerical integrator for general systems such as a Runge-Kutta or predictor-corrector method. 

Our discussion so far has considered the use of SHAKE with the Verlet algorithm Versions have also been derived for other integration schemes, such as the leap-froj algorithm, the predictor-corrector methods and the velocity Verlet algorithm. In the cast of the velocity Verlet algorithm, the method has been named RATTLE [Anderson 1983]... 

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... 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

The temperature, pore width and average pore densities were the same as those used by Snook and van Megen In their Monte Carlo simulations, which were performed for a constant chemical potential (12.). Periodic boundary conditions were used In the y and z directions. The periodic length was chosen to be twice r. Newton s equations of motion were solved using the predictor-corrector method developed by Beeman (14). The local fluid density was computed form... 

M71 Solution of ordinary differential equations predictor-corrector method of Milne 7100 7188... 

The multistep method (5.25) is explicit if bQ = 0, otherwise it is implicit. These latter are the best ones due to their improved stability properties. To use an implicit formula, however, we need an initial estimate of yi+1. The basic idea of the predictor - corrector methods is to estimate y1+1 by a p-th order explicit formula, called predictor, and then to refine yi+1 by a p-th order implicit formula, which is said to be the corrector. 

The great advantage of the predictor - corrector methods is that in addition to y1+, in expression (5.25) we need only previously computed (and saved) function values. Thus, the computational cost depends on the number of corrections and does not depend on the order p of the particular formula. 

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. 

In the predictor - corrector methods the magnitude of the first correction is an immediate error estimate with no additional cost. 

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. 

If j30 = 0, the method is explicit and the computation of is straightforward. If 30 + 0, the method is implicit because an implicit algebraic equation is to be solved. Usually, two algorithms, a first one explicit and called the predictor, and a second one implicit and called the corrector, are used simultaneously. The global method is called a predictor-corrector method as, for example, the classical fourth-order Adams method, viz. 

R. M. Thomas and T. E. Simos, A family of hybrid exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrodinger equation, J. Comput. Appl. Math., 1997, 87, 215-226. 

G. Psihoyios and T. E. Simos, The numerical solution of the radial Schrodinger equation via a trigonometrically fitted family of seventh algebraic order Predictor-Corrector methods, J. Math. Chem., 2006, 40(3), 269-293. 

Predictor-Corrector methods have been constructed attempting to combine the best properties of the explicit and implicit methods. The multistep methods are using information at more than two points. The additional points are ones at which data has already been computed. In one view, Adams methods arise from underlying quadrature formulas that use data outside of specifically approximate solutions computed prior to t . 

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 ... 

These results are similar to that from the energy balance. The differences are the result of round off errors in the simple finite difference calculation scheme used here (i.e., more complicated predictor-corrector methods would yield more accurate results ). 

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