# 1.8 Multi-Step Methods

Adams-Moulton methods are implicit methods of the form,

 $v^{n+1} - v^{n} = {\Delta t}\sum _{i=0}^ s \beta _ i f^{n+1-i}.$ (1.137)

These methods use the same time derivative approximation as the Adams-Bashforth methods, however they include the $$n+1$$ value of $$f$$.

 $$p$$ $$\beta _0$$ $$\beta _1$$ $$\beta _2$$ $$\beta _3$$ 1 1 2 $$\frac{1}{2}$$ $$\frac{1}{2}$$ 3 $$\frac{5}{12}$$ $$\frac{8}{12}$$ $$-\frac{1}{12}$$ 4 $$\frac{9}{24}$$ $$\frac{19}{24}$$ $$-\frac{5}{24}$$ $$\frac{1}{24}$$
Table 3: Coefficients for Adams-Moulton methods. Note: the $$p=1$$ method is the backward Euler method, and the $$p=2$$ method is the Trapezoidal method.

The coefficients for the first through fourth order methods are given in the table above. Figure 1.20: Adams-Moulton stability regions for $$p=1$$ through $$p=4$$ methods. Note: $$p=1$$ is stable outside of contour, the $$p=2$$ integrator is stable in the left-half plane, and $$p\geq 3$$ are stable inside their respective contours.

The stability boundary for these methods are shown in Figure 1.20. While the stability regions are larger than the Adams-Bashforth methods, for $$p>2$$ the methods have bounded stability regions. Thus, they will not be appropriate for stiff problems.