the Adams-Moulton method
/ Adams-Moulton (AD-amz MOLE-tun) /
Adams-Bashforth fits a polynomial through PAST slopes and extends it forward — but extending a polynomial beyond its data (extrapolation) is always shaky. Adams-Moulton makes one improvement that changes everything: it lets the polynomial pass through the slope at the NEW point too, not just past ones. Now you are interpolating, fitting a curve within its data rather than guessing beyond it, which is far more accurate and stable. The price is that the new slope depends on the new value you are trying to find — so the method is implicit.
Precisely, the formula y_(n+1) = y_n + h * (weighted sum of past slopes AND f(t_(n+1), y_(n+1))) includes the not-yet-known slope at the new point. The simplest case is the trapezoidal rule, y_(n+1) = y_n + (h/2)(f_n + f_(n+1)); a higher member is the three-step Adams-Moulton, fourth order. Because y_(n+1) sits inside the right-hand side through that f(t_(n+1), y_(n+1)) term, you cannot read off the answer — you must solve an algebraic equation for it each step (by Newton's method if f is nonlinear, or by predictor-corrector iteration).
What you buy with that extra trouble is real. For the same number of steps, Adams-Moulton is more accurate than Adams-Bashforth of comparable order, and crucially it has a much larger region of absolute stability — implicit methods stay stable where explicit ones blow up, which is exactly what stiff problems demand. In practice Adams-Bashforth (the predictor) and Adams-Moulton (the corrector) are run together as a predictor-corrector pair: the explicit method guesses, the implicit formula refines, and you get implicit-quality accuracy without solving a full nonlinear equation from scratch.
The trapezoidal rule y_(n+1) = y_n + (h/2)(f_n + f(t_(n+1), y_(n+1))) is the simplest Adams-Moulton method. The unknown y_(n+1) appears on both sides, so for nonlinear f you solve for it each step, or feed in an Adams-Bashforth prediction and correct once.
Includes the new slope too — implicit, more accurate and far more stable.
Being implicit does not by itself make Adams-Moulton good for stiff problems of every order: the high-order Adams-Moulton methods have limited stability regions. For genuine stiffness the BDF family is usually preferred.