Page 97 - Standard Handbook Of Petroleum & Natural Gas Engineering

86 Mathematics

Ei+l - Yi+l - Yicl
2-k - 1

where k is the order o the expression (e.g., k = 4 for the foregoing Runge-
Kutta formula). The step size can be adjusted to keep the error E below some
predetermined value.
The Adams Open Formulas are a class of multistep formulas such that the first-
order formula reproduces the Euler formula. The second-order Adams open
formula is given by

yitl = yi +Ax -fi --fi&,]+O(Ax)'
3
1
[ 2 2
This formula and the higher-order formulas are not self starting since they
require fi-l, fi-,, etc. The common practice is to employ a Runge-Kutta formula
of the same order to compute the first term(s) of y,. The general Adams open
formula may be written as

and the coefficients p are given in Table 1-17 for n = 0,1, . . .,5.
Adams Closed Formulas require an iterative method to solve for Y,+~, since the
right side of the expression requires a value for fi+,. The iteration of estimating
y, evaluating f, and obtaining a new estimate of y is repeated until it converges
to the desired accuracy. The general formula is

and the coefficients p* are given in Table 1-18.

Table 1-17
Coefficients S,, of the Open Adams Formulas
k
n 0 1 2 3 4 5 o*
~ ~
0 1 1
1 312 -1 12 2
2 2311 2 -1 611 2 511 2 3
3 55/24 -59124 37/24 -9124 4
4 1901/720 -2774/720 261 61720 -1 2741720 251 1720 5
5 427711 440 -792311 440 99a211440 -729811 440 287711 440 -47511 440 6
0 is the order of the method