A fixed time step spends the same computational effort everywhere, even when the solution changes rapidly in one region and slowly in another.

An adaptive ODE solver estimates the local error of a proposed step and adjusts the step size to keep that error near a requested tolerance.

If the estimated error is too large, the step is rejected or repeated with a smaller $h$. If the error is much smaller than necessary, a larger next step can reduce computational cost.

Embedded Runge-Kutta pairs efficiently obtain two approximations of different orders from largely shared derivative evaluations. Their difference provides a local error estimate.

Adaptive stepping does not remove numerical error; it allocates resolution where the evolving solution requires it and makes the accuracy target explicit.