(21) |

Adams methods are based on the idea of approximating the integrand with a polynomial within the interval (

The first order AB and AM methods are simply the forward and the backward Euler methods respectively. The second order
versions (obtained by using a linear interpolant)
of these methods are quite popular. The second order Adams-Bashforth (AB2) method is given by

(22) |

Note that the AB2 method is explicit and hence only conditionally stable. Moreover, the AB2 method requires the solution from the

The second order Adams-Moulton (AM2) is an implicit technique, sometimes referred to as the *trapezoidal* rule.
The time-stepping equation for AM2 is given by

(23) |

The implicit nature of the method is evident from Eq. 24. For a non-linear IVP, we have to solve a non-linear algebraic equation at every time step. This is much more expensive as compared to the explicit AB2 method. However, being an implicit technique, AM2 does not suffer from the numerical instability of the AB2 for relatively large values of the time step. Once again, it is a trade-off between stability and computational cost, since both AM2 and AB2 are second order accurate.