Adams-Moulton fourth step method

Adams-Moulton Fourth-Step Method Predictor... 

The predictor calls for four previous values in Adams-Moulton and Milne s algorithms. We obtain these by the fourth-order Runge-Kutta method. Also, we can reduce the step size to improve the accuracy of these methods. Milne s method is unstable in certain cases because the errors do not approach zero as we reduce the step size, h. Because of this instability, the method of Adams-Moulton is more widely used. 

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. 

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 numerical integration of the equations of motion, equations (4) and (5), is straightforward. Various well known integration methods are used. For example, the fixed step-size fourth-order Runge-Kutta and Adams-Moulton fifth-order predictor/sixth-order corrector (initiated with a fourth-order Runge-Kutta) algorithms (see, for example. Ref. 7h) are often used. 

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... 

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