Stiff equation

A stiff differential equation contains rapidly decaying modes alongside much slower dynamics. An explicit method may then require a small step for stability even when accuracy only requires resolving the slow modes.

If a linear system has negative eigenvalues with widely separated magnitudes, an explicit method can be forced to resolve the fastest decaying mode solely for stability. For eigenvalues $-1$ and $-100$, forward Euler requires $h\le 0.02$, whereas backward Euler is stable for every positive step size.