Stiff problems and stability
What makes an ODE stiff, why explicit methods become unstable with few points, and why implicit, low-order and adaptive-step methods are preferred.
What a stiff equation is
An ODE is stiff when its solution mixes very different time scales: a very fast transient followed by slow, stable behaviour. With a constant step, an explicit method needs a great many points to avoid becoming unstable in the transient, even though the slow part does not need them. The model problem analysis of quantifies the phenomenon: the explicit method's step is limited by , while the implicit one is stable at any step.