predictor-corrector method

Take a rough step with an explicit formula, then use that rough answer to plug into a MORE ACCURATE (but otherwise unusable) implicit formula and refine it.

A predictor-corrector method pairs an explicit multistep formula (the predictor, e.g. Adams-Bashforth) with an implicit one of the same or higher order (the corrector, e.g. Adams-Moulton): the predictor produces a first estimate of $y_{n+1}$ , $f$ is evaluated there, and that value substitutes for the corrector's own unknown $f_{n+1}$ term -- turning an equation that would otherwise need to be solved for $y_{n+1}$ into one that can be evaluated directly. This PECE (Predict-Evaluate-Correct-Evaluate) pattern gets most of an implicit method's accuracy without ever solving an implicit equation.

Examples

PECE in practice: Predict $y_{n+1}$ with Adams-Bashforth, Evaluate $f$ there, Correct with Adams-Moulton using that value in place of $f_{n+1}$ , Evaluate $f$ again at the corrected value.