I've studied various numerical methods in my scientific computing class. These include:
(1) Runge-Kutta
(2) Multistep Backward Difference
(3) Multistep Adams Bashforth
(4) Multistep Adams Moulton
But although the book mentions all of them, there is scarce reference to the last one, Adams Moulton, so I'm not familiar what functions its best at approximating. When would I need to use it? Thanks!
AB and AM usually are not used because they are not A-stable. That's could be the reason.
Actually, if you use the following numerical method $y_{n+2} - 3y_{n+1}+2y_n=\frac{k}{12}(-5f(t_n,y_n) - 20f(t_{n+1},y_{n+1}) +13f(t_{n+2},y_{n+2}))$ to the test problem
$\begin{cases} y'(t)=0 \quad t \in (0,\bar{t}] \\ y(0)=y_0 \end{cases}$
the numerical solution diverges. If you take $y_1=y_0 + \mathcal{O(k^2)}$ (this method is of second order), then $y_2=y_0+3ck^2, \dots ,y_n=y_0 + c(2^m-1) \frac{\bar{t}}{m}^2$ [Here $m=\frac{\bar{t}}{k}$]. Taking the limit as $k \rightarrow 0$ you get that $y_n \rightarrow \infty$