the region of absolute stability
Accuracy is about how close you are to the truth; stability is about whether your computation behaves at all or runs away to nonsense. Even on a problem whose true solution quietly decays to zero, a numerical method with too large a step can produce a result that grows, oscillates, and blows up — pure numerical garbage with no relation to reality. The region of absolute stability is the precise map of which step sizes keep a method well-behaved, and which let it explode.
Precisely, you test every method on one benchmark equation, y' = lambda y, whose exact solution e^(lambda t) decays when the real part of lambda is negative. Applying a method gives y_(n+1) = R(z) * y_n where z = h * lambda and R is a function fixed by the method. The numerical solution stays bounded (matching the decaying truth) exactly when the magnitude of R(z) is at most 1. The set of complex z = h*lambda for which that holds is the region of absolute stability — a shape in the complex plane. For a given lambda you must choose h so that h*lambda lands inside that region.
This shape is the whole story of stiffness. Explicit methods like Euler and RK4 have SMALL, bounded stability regions, so a strongly negative lambda forces h to be tiny just to stay inside — which is exactly the curse of stiffness. Implicit methods like backward Euler have HUGE stability regions (backward Euler is stable for the entire left half-plane, called A-stability), so they take large steps on stiff problems without exploding. Reading a method's stability region tells you, before you compute anything, how large a step it can survive — and whether it can handle stiffness at all.
Forward Euler applied to y' = lambda y gives y_(n+1) = (1 + z) y_n with z = h*lambda; stability needs |1 + z| <= 1, a unit disk centred at -1. For real lambda = -100 this demands h <= 0.02. Backward Euler gives y_(n+1) = y_n / (1 - z), stable for the entire left half-plane — any h works.
Stability needs h*lambda inside the region — small for explicit, vast for implicit.
Absolute stability is separate from accuracy and from convergence. A step inside the stability region only guarantees the solution stays bounded, not that it is accurate; and stability for a fixed h is a different question from convergence as h -> 0.