1550078481-Ordinary_Differential_Equations__Roberts_

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

1. 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].