1.8 Multi-Step Methods

1.8.2 Adams-Moulton Methods

Measurable Outcome 1.12, Measurable Outcome 1.15, Measurable Outcome 1.17, Measurable Outcome 1.18, Measurable Outcome 1.19

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.

The graph shows the shrinking, mostly non-overlapping, Adams-Moulton stability regions for p=1 through p=4 methods.
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.