Adams-moulton Methods

Using (1.2), we obtain the error term as

........................1.5

where,

Since g(s) does not change sign in [0, 1], we get by the mean value theorem

Note :
From (1.5), we obtain that the truncation error is of order O(hk 2). Therefore, a k-step Adams-Moulton method is of order k 1.

By choosing different values for k, we get different methods.

for k = 0 : We get the method

..................1.6

which is the backward Euler’s method. Using (1.5), we obtain the error term as

Therefore, the method is of first order.
k = 1: We get the method

......................1.7

This is also a single step method and we do not require any starting values. This method is also called the trapezium method.

Using (1.5), we obtain the error term as

Therefore, the method is of second order.
k = 2: We get the method

.....................1.8