Stability of an ODE

An ODE is stable if small perturbations in the initial condition remain bounded for all future time.

Formally, the ODE $\dot{y}=g(t,y)$ is stable if: for any $\epsilon >0$ there exists $\delta >0$ such that

$$|y_A(t_0)-y_B(t_0)|<\delta ⟹|y_A(t)-y_B(t)|<\epsilon \text{for all }t>t_0$$

where $y_A$ and $y_B$ are two solutions starting from nearby initial conditions.

In other words, two solutions that start close together do not diverge from each other. This is a property of the ODE itself, independent of the numerical method used to solve it.

The stability of the ODE is a prerequisite for the Lax equivalence theorem: we need the problem to be well-posed before asking whether a numerical method converges. The stability of the numerical method itself is a separate question - see Stability of a method.

Stability of a method | Convergence | Lax Equivalence Theorem