Corrector Adams method

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. 

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 . 

The well-known predictor-corrector Adams-Bashforth method of algebraic order four. 

Figure 5.8 Stability region in the complex plane for the modified Euler (Euler predictor-corrector), Adams, and Adams-Moulton methods.

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... 

When the quadrature of eq 2 cannot be performed analytically the integration should be carried out numerically by robust routines such as the Runge-Kutta, Adams-Moulton predictor-corrector or Bulirsch-Stoer methods with step size and error control [53, 55, 56], These routines can also be found in computer codings at Netlib and in standard books on computer codes [53]. 

An extension of the multistep methods is the predictor-corrector approach. Here, the Adams-Bashford equation may be used to calculate a predicted value for y at (x + h). Then a second, corrector, equation is used to refine the valueof y. If the difference between the predictor and corrector values is within specified error limits, thecalculation is continued to the next step, otherwise the step size will be adjusted to maintain the error limits specified. With fewer function evaluations per step, these methods can be faster than the Runge-Kutta methods however, they are not self-starting. 

In 26 the authors have developed a new trigonometrically-fitted predictor-corrector (P-C) scheme based on the Adams-Bashforth-Moulton P-C methods. In particular, the predictor is based on the fifth algebraic order Adams-Bashforth scheme and the corrector on the sixth algebraic order Adams-Moulton scheme. More specifically the new developed scheme integrates exactly any linear combination of the functions ... 

In 30 the authors have developed trigonometrically fitted Adams-Bashforth-Moulton predictor-corrector (P-C) methods. It is the first time in the literature that these methods are applied for the efficient solution of the resonance problem of the Schrodinger equation. The new trigonometrically fitted P-C schemes are based on the well known Adams-Bashforth-Moulton methods. In particular, they are based on the fourth order Adams-Bashforth scheme (as predictor) and on the fifth order Adams-Moulton scheme (as corrector). More... 

In 37 the authors have developed trigonometrically fitted Adams-Bashforth-Moulton predictor-corrector (P-C) methods. The new trigonometrically fitted P-C schemes are based on the well known Adams-Bashforth-Moulton... 

The Adam-Bashforth methods are frequently used as predictors and the Adam-Moulton methods are often used as correctors. The combination of the two formulas results in predictor-corrector schemes. 

RFPLO code, a predictor corrector method of Adams-Bashford-Moulton type is used. The eigenvalues are found by matching the inward and outward solutions. 

Since the predictor-corrector method is particularly suitable (when it works) in terms of computational times and memory allocation (it does not need to store the Jacobian), it is used with nonstiff problems and with algorithms that are not good at solving stiff problems, but with better accuracy featiu-es (usually the Adams-Moulton methods are adopted). 

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

One of the most common numerical methods used in molecular dynamics to solve Newton s equations of motions is the Velocity Verlet integrator. This is typically implemented as a second order method, and we find that it can become numerically tmstable dtuing the course of hyperthermal collision events, where the atom velocities are often far from equilibrium. As an alternative, we have implemented a fifth/ sixth order predictor-corrector scheme for our calculations. Specifically, the driver we chose utilizes the Adams-Bashforth predictor method together with the Adams-Moulton corrector method for approximating the solution to the equations of motion. 

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]. 

Use the Adams-Moulton predictor-corrector method to approximate... 

Using the Runge-Kutta order four method for starting, the Adams-Bashforth method as the predictor, and the Adams-Moulton method as a corrector, approximate the solution to... 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

The Adams-Mouhon method, being a predictor-corrector scheme, involved the use of the corrector formula until the difference in two successive iterations was below a predetermined value, but in most cases it only used the corrector formula once. 

Ccxnputation times also follow the expected evolution, increasing with the niunber of function evaluations per step required by the method. The Adams-Moulton scheme, however, which requires only two e uations per step, like the modified Euler scheme, was only slightly faster than the fourth-order Runge-Kutta due to the amount of computation involved in the predictor and corrector formulas. The former also provided extremely low errors, when applicable, but showed a tendency to become unstable at higher step sizes. 

The quasiclassical trajectory method was used to study this system, and the variable step size modified Bulirsch-Stoer algorithm was specially developed for recombination problems such as this one. Comparisons were made with the fourth order Adams-Bashforth-Moulton predictor-corrector algorithm, and the modified Bulirsch-Stoer method was always more efficient, with the relative efficiency of the Bulirsch-Stoer method increasing as the desired accuracy increased. We measure the accuracy by computing the rms relative difference between the initial coordinates and momenta and their back-integrated values. For example, for a rms relative difference of 0.01, the ratio of the CPU times for the two methods was 1.6, for a rms relative difference of 0.001 it was 2.0, and for a rms relative difference of 10 it was 3.3. Another advantage of the variable step size method is that the errors in individual trajectories are more similar, e.g. a test run of ten trajectories yielded rms errors which differed by a factor of 53 when using the modified Bulirsch-Stoer... 

The implicit multistep methods add stability but require more computation to evaluate the implicit part. In addition, the error coefficient of the Adams-Moulton method of order k is smaller than that of the Adams Bashforth method of the same order. As a consequence, the implicit methods should give improved accuracy. In fact, the error coefficient for the imphcit fourth-order Adams Moulton method is 19/720, and for the explicit fourth-order Adams Bashforth method it is 251/720. The difference is thus about an order of magnitude. Pairs of exphcit and implicit multistep methods of the same order are therefore often used as predictor-corrector pairs. In this case, the explicit method is used to calculate the solution,, at v +i. Furthermore, the imphcit method (corrector) uses y + to calculate /(x +i,y +i), which replaces /(x +i,y +i). This allows the solution, y +i, to be improved using the implicit method. The combination of the Adams Bashforth and the Adams Moulton methods as predictor orrector pairs is implemented in some ODE solvers. The Matlab odel 13 solver is an example of a variable-order Adams Bashforth Moulton multistep solver. 

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