For the test problem dy/dt = λy with Re(λ) < 0, a numerical method is stable if |y_{n+1}| ≤ |y_n|. The stability region is the set of hλ values for which the method is stable. An A-stable method (like implicit Euler or Crank-Nicolson) is stable for all hλ with Re(hλ) < 0, making it safe for stiff problems. Explicit methods have bounded stability regions, severely limiting step size for stiff systems.
From your study of stiff differential equations, you know that stiffness creates a fundamental tension: the exact solution may vary slowly and smoothly, yet the ODE has eigenvalues with large negative real parts that force explicit methods to take absurdly small steps. The concept of stability regions makes this tension precise. The standard test problem is the scalar equation dy/dt = λy with Re(λ) < 0, whose exact solution y(t) = y₀e^{λt} decays exponentially. A numerical method is stable for this problem if the numerical approximation also decays — that is, if |yₙ₊₁| ≤ |yₙ|. The stability region is the set of values hλ in the complex plane for which this decay condition holds.
For forward (explicit) Euler, applying one step gives yₙ₊₁ = (1 + hλ)yₙ. The method is stable when |1 + hλ| ≤ 1, which describes a disk of radius 1 centered at −1 in the complex hλ-plane. If λ = −1000 (a stiff decay rate) and you want hλ inside this disk, you need roughly h < 2/1000 = 0.002. This is a stability restriction, not an accuracy restriction — the solution is nearly constant and large steps would be perfectly accurate, but the method blows up unless h is tiny. For backward (implicit) Euler, yₙ₊₁ = yₙ/(1 − hλ), and the stability region is everything outside the unit disk centered at 1 — which includes the entire left half-plane. No step-size restriction is needed for stability when Re(λ) < 0.
A method whose stability region contains the entire left half of the complex plane — all hλ with Re(hλ) < 0 — is called A-stable. The backward Euler method and the Crank-Nicolson (trapezoidal) method are both A-stable. For stiff problems, A-stability is the critical property: it means you can choose h based purely on accuracy requirements, without worrying that the method will blow up. The Dahlquist barrier theorem shows that no explicit linear multistep method can be A-stable, and among implicit linear multistep methods, the highest-order A-stable method is the trapezoidal rule (order 2). This is why implicit methods dominate stiff ODE solving in practice.
The conceptual separation of stability from accuracy is the key insight. Accuracy is about truncation error — how well the method approximates the true derivative. Stability is about error propagation — whether small perturbations grow or decay as the method iterates. A method can be highly accurate (high order) yet unstable for a particular hλ, producing catastrophic blowup. Conversely, backward Euler is only first-order accurate but is unconditionally stable for decaying problems. In practice, stiff ODE solvers use high-order implicit methods (like BDF methods or implicit Runge-Kutta) that combine accuracy with large stability regions, allowing efficient integration of systems where eigenvalues span many orders of magnitude.