adaptive step size

Letting a numerical method choose its own step size as it runs, shrinking h where the solution changes fast and growing it where the solution is calm.

An adaptive step-size method estimates its own local truncation error at every step (typically by comparing two embedded estimates of different order, as in the Runge-Kutta-Fehlberg method) and uses that estimate to accept or reject the step and to rescale $h$ for the next one -- spending small steps only where the solution actually demands them, rather than using one fixed $h$ for the whole interval.

Examples

A solution that's nearly flat for most of the interval but spikes sharply near the end lets an adaptive method take large steps early and shrink $h$ automatically only near the spike.