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Finally, we replace $(tn+1) and $(tn) by yn+1 and yn, respectively, and carry out some algebraic simplification. For a constant step size h we obtain
Equation (5) is the second order Adams-Bashforth formula. It is an explicit formula
We note in passing that the first order Adams-Bashforth formula, based on the polynomial P1(t) = fn of degree zero, is just the original Euler formula.
More accurate Adams formulas can be obtained by following the procedure outlined above, but using a higher degree polynomial and correspondingly more data points. For example, suppose that a polynomial P4(t) of degree three is used. The coefficients are determined from the four points (tn, yn), (tn—1, yn—1), (tn—2, yn—2), and (t^, yn—3). Substituting this polynomial for $( t) in Eq. (2), evaluating the integral, and simplifying the result, we eventually obtain the fourth order Adams-Bashforth formula, namely,
A variation on the derivation of the Adams-Bashforth formulas gives another set of formulas called the Adams-Moulton4 formulas. To see the difference, let us again consider the second order case. Again we use a first degree polynomial Q2(t ) = at + 3,
Solving for A and B, we obtain
Replacing $(t) by P2(t) and evaluating the integral in Eq. (2), we find that
for yn+1 in terms of yn and yn_ 1 and has a local truncation error proportional to h3.
4Forest Ray Moulton (1872-1952) was an American astronomer and administrator of science. While calculating ballistics trajectories during World War I, he devised substantial improvements in the Adams formula.
8.4 Multistep Methods
but we determine the coefficients by using the points (tn, yn) and (tn+1; yn+j). Thus a and 3 must satisfy
atn + 3 = fn ’ n n (7)
atn+l + 3 = fn+1,
and it follows that
f — f f t — f t
a = _n+i------n, 3 = _2_2+i n+y_n. (8)
h H h w
Substituting Q2(t) for 0'(t) in Eq. (2) and simplifying, we obtain
yn+l = yn + 1 hfn + 1 hf(tn+V yn+1)> (9)
which is the second order Adams-Moulton formula. We have written f (tn+ j; yn+1 ) 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
yn+1 = yn + (h/24)(9 fn+1 + 19 n - 5 fn-1 + fn-2). (10)
Observe that this is also an implicit formula because yn+1 appears in fn+1.
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
yn^ yn^ yn-v and yn are kn°w^ we can compute fn^ fn_x, and fn, ami
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+1. We can, of course, continue to use the corrector formula (10) if the change in yn+1 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 y1 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.