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f — f f t — f t
a = _n+i-----n, â = _n_n±i------------n+l_n. (8)
h H h w
Substituting Q2(t) for ô'(t) in Eq. (2) and simplifying, we obtain
Óï+1 = Óï + 1 hfn + 1 hf (tn+V Óï+1)> (9)
which is the second order Adams-Moulton formula. We have written f (tn+ ð yn+ x) in the last term to emphasize that the Adams-Moulton formula is implicit, rather than explicit, since the unknown yn 1 appears on both sides of the equation. The local truncation error for the second order Adams-Moulton formula is proportional to h3.
The first order Adams-Moulton formula is just the backward Euler formula, as you might anticipate by analogy with the first order Adams-Bashforth formula.
More accurate higher order formulas can be obtained by using an approximating polynomial of higher degree. The fourth order Adams-Moulton formula, with a local truncation error proportional to h5, is
Óï+1 = Óï + (h/24)(9 fn+1 + 19 Ï — 5 fn—1 + fn—2). (j0)
Observe that this is also an implicit formula because yn+x appears in fn+x.
Although both the Adams-Bashforth and Adams-Moulton formulas of the same order have local truncation errors proportional to the same power of h, the Adams-Moulton formulas of moderate order are in fact considerably more accurate. For example, for the fourth order formulas (6) and (10), the proportionality constant for the Adams-Moulton formula is less than 1/10 of the proportionality constant for the Adams-Bashforth formula. Thus the question arises: Should one use the explicit (and faster) Adams-Bashforth formula, or the more accurate but implicit (and slower) Adams-Moulton formula? The answer depends on whether by using the more accurate formula one can increase the step size, and therefore reduce the number of steps, enough to compensate for the additional computations required at each step.
In fact, numerical analysts have attempted to achieve both simplicity and accuracy by combining the two formulas in what is called a predictor-corrector method. Once
Óï—3, Óï—2, Óï—^ and Óï are kn°w^ we can compute fn—3, ^—^ fn—v and fn, ÇØ² then use the Adams-Bashforth (predictor) formula (6) to obtain a first value for yn 1. Then we compute fn+1 and use the Adams-Moulton (corrector) formula (10), which is no longer implicit, to obtain an improved value of yn+r We can, of course, continue to use the corrector formula (10) if the change in yn+x is too large. However, if it is necessary to use the corrector formula more than once or perhaps twice, it means that the step size h is too large and should be reduced.
In order to use any of the multistep methods it is necessary first to calculate a few y. by some other method. For example, the fourth order Adams-Moulton method requires values for ó and y2, while the fourth order Adams-Bashforth method also requires a value for y3. One way to proceed is to use a one-step method of comparable accuracy to calculate the necessary starting values. Thus, for a fourth order multistep method,
Chapter 8. Numerical Methods
one might use the fourth order Runge-Kutta method to calculate the starting values. This is the method used in the next example.
Another approach is to use a low order method with a very small h to calculate y1, and then to increase gradually both the order and the step size until enough starting values have been determined.
Consider again the initial value problem
Ó = 1 - t + 4ó, ó(0) = 1. (11)
With a step size of h = 0.1 determine an approximate value of the solution ó = ô() at t = 0.4 using the fourth order Adams-Bashforth formula, the fourth order Adams-Moulton formula, and the predictor-corrector method.
For starting data we use the values of yv y2, and y3 found from the Runge-Kutta method. These are tabulated in Table 8.3.1. Next, calculating the corresponding values of f ( t, y), we obtain
Ó0 = 1 f0 = 5
y1 = 1.6089333, fj = 7.3357332,
y2 = 2.5050062, f2 = 10.820025,
y3 = 3.8294145, f, = 16.017658.
Then from the Adams-Bashforth formula, Eq. (6), we find that y4 = 5.7836305. The exact value of the solution at t = 0.4, correct through eight digits, is 5.7942260, so the error is -0.010595.
The Adams-Moulton formula, Eq. (10), leads to the equation
y4 = 4.9251275 + 0.15y4,
from which it follows that y4 = 5.7942676 with an error of only 0.0000416.
Finally, using the result from the Adams-Bashforth formula as a predicted value of ô(0.4), we can then use Eq. (10) as a corrector. Corresponding to the predicted value of y4 we find that f4 = 23.734522. Hence, from Eq. (10), the corrected value of y4 is 5.7926721. This result is in error by -0.0015539.