the Adams-Bashforth method

Runge-Kutta methods get high accuracy by sampling the slope several times within each step — but those samples are computed and then thrown away. Multistep methods take a thriftier view: we already computed the slope at the last few steps, so why not REUSE that history? Adams-Bashforth is the explicit multistep family. It fits a polynomial through the last several known slopes, extends that polynomial across the new step, and integrates under it to leap forward — getting high order with just ONE new slope evaluation per step.

Precisely, you keep the recent slope values f_n, f_(n-1), f_(n-2), ... and form y_(n+1) = y_n + h * (a weighted sum of those past slopes). The weights come from integrating the interpolating polynomial that passes through the stored slopes. For example, the two-step Adams-Bashforth method is y_(n+1) = y_n + (h/2)(3 f_n - f_(n-1)), which is second order; the popular four-step version uses the last four slopes and is fourth order. It is explicit because every slope it needs is from the past — the new value y_(n+1) never appears on the right-hand side, so you just compute it.

The win is efficiency: where RK4 needs four slope evaluations per step, four-step Adams-Bashforth needs only one (the rest are recycled), so when each f evaluation is expensive, multistep methods can be much cheaper for the same order. The catch is they are not self-starting — at the very beginning you have no history, so you must launch with a one-step method like RK4 to build up the first few values. And being explicit, plain Adams-Bashforth shares RK4's weakness on stiff problems, which is why it is usually paired with its implicit cousin Adams-Moulton as a predictor-corrector.