Page 98 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 87

Table 1-18
Coefficients p*,, of the Closed Adam Formulas
k
n 0 1 2 3 4 5 (7'
0 1 1
1 1 12 112 2
2 511 2 at1 2 -1 I1 2 3
3 9/24 19124 -5124 1124 4
4 2511720 646l720 -264l720 1061720 -191720 5
5 47511 440 142711 440 -79a11440 48211 440 -1 7311 440 2711 440 6
0 is the order of the method

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 yi+l, 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

1. Predictor

4
Yi+l = yi-5 + - W2f - f i-1 + 2f i-*)
3
2. Corrector

1
yj+, = -wfi+l +4fi +fi&,)
3
although the Milne method, like the Adams formulas, is not self starting.

The Humming method [12] applies a predictor yo, then a modifier i" which
provides a correction for the estimate of error in the predictor and corrector,
and then iterates the corrector y" as desired. The procedure is

1. Predictor

4
3
Y!?, = y,-,+-k(2fi -f,4 +2f,_,)
2. Modifier

(0) 112
(Y,
F!",: = Yi+l + - - Y1O')
121