A time-stepping method can have small local truncation error and still produce a useless solution if numerical perturbations grow from step to step.

For the test equation

$$y'=\lambda y,$$

a one-step method often produces an update

$$y_{n+1}=R(h\lambda)y_n,$$

where $R$ is the method's amplification factor.

When the true solution should decay, numerical stability requires the discrete updates to decay as well. A common condition is

$$|R(h\lambda)|\le1.$$

The set of values of $h\lambda$ satisfying this condition is the method's stability region.

Stability of a time integrator concerns error growth across repeated evolution steps. It is different from backward stability in floating-point algorithms, although both ask whether numerical perturbations are amplified.