The Initial Value Problem y' = f(x, y); y(c) = d 101
form= 1- is known as the midpoint rule and has a local discretization error
of h^3 y(^3 ) (()/6, where ( E (xn-l, Xn+ 1 ). The midpoint rule has the simplicity
of Euler's method and has a smaller lo cal error; however, it requires one
starting value and is generally less stable than Euler's method. (The term
"stable" is discussed in section 2.4.3.)
2.4.2.3 Adams-Moulton Methods
The Adams-Bashforth methods and the Nystrom methods employ polyno-
mials which interpolate f at Xn and the preceding points Xn-1, ... , Xn-m·
If we find an interpolating polynomial which interpolates f at Xn-l, Xn, ... ,
Xn-m and integrate from Xn to Xn+l, we obtain a set of formulas known as
Adams-Moulton formulas. In our notation r = n + 1, p = 0, and q = 1. The
first few Adams-Moulton formulas and their associated local discretization
errors follow.
m Adams-Moulton formulas Error
0
_ + h(f n+l + f n) h3y(3) (()
Yn+l -Yn 2 12
1
h(5j n+l + 8j n - f n-i)
Yn+l = Yn + 12
h4y(4) (()
24
2
h(9fn+l + 19fn - 5fn-l + fn-2)
Yn+l = Yn + 12
19h^5 y<^5 l(()
720
In each case ( E (xm-n, Xn+i)· All of these formulas are implicit formu-
las for Yn+i, since Yn+l appears on the right-hand side of each equation in
fn+i = f(xn+l, Yn+ 1 ). The formula for m = 0 is called the trapezoidal
scheme. Forest Ray Moulton (1872-1952) was an American astronomer who
improved the Adams formula substantially while computing numerical so lu-
tions to ballistic problems during World War I.
EXERCISES 2.4.2
- a. Compute a numerical approximation to the initial value problem
y' = x^2 - y ; y(O) = 1 on the interval [O, l] using the Adams-
Bashforth formula form= 1. Use a constant stepsize h = .1 and
use the exact solution values for starting values.
b. Estimate the maximum local discretization error on the interval
[O, 1].