next up previous

2.3.1 The Adams-Bashforth and Adams-Moulton Formulas     continued...

Note that for a formula of order p, both the Adams-Bashforth and Adams-Moulton formulas interpolate the function f on p t-points. The t-points overlap and are illustrated graphically below in Figure 7.

The lowest order Adams-Moulton formula involves interpolating the single value and an easy calculation leads to the formula

which defines implicitly. The resulting formula is called the backward Euler formula. From its definition it is clear that it has the same accuracy as the forward Euler method; its advantage is vastly superior stability. The second order Adams-Moulton method also does not use previously computed solution values; it is called the trapezoidal rule because it generalizes the trapezoidal rule for integrals to differential equations:

The third order formula is more typical because it does involve a previously computed value. When the step size is a constant h, it is