AStabilität

AStabilität, or A-stability, is a stability concept used for time integration methods in the solution of ordinary differential equations, especially stiff systems. It focuses on how numerical schemes handle exponential decay that arises in stiff problems.

Mathematical definition: Consider the linear test equation y′ = λ y with λ in the complex plane and Re(λ) ≤ 0. For a fixed step size h > 0, a numerical method produces a recurrence y_{n+1} = R(hλ) y_n, where R is the method’s stability function. The method is A-stable if the entire left half-plane Re(hλ) ≤ 0 lies inside its stability region, i.e., |R(z)| ≤ 1 for all z with Re(z) ≤ 0. In other words, the method remains non-amplifying for all decaying linear modes.

Typical examples: Implicit Euler has R(z) = 1/(1 − z) and is A-stable (in fact, L-stable, meaning it damps stiff modes as z → −∞). The trapezoidal rule (Crank-Nicolson) has R(z) = (1 + z/2)/(1 − z/2) and is A-stable but not L-stable. Gauss-Legendre implicit Runge-Kutta methods are A-stable, offering high accuracy for stiff problems, while Radau IIA methods are also A-stable and are often L-stable, making them particularly robust for stiff dynamics.

Applications and implications: A-stability is desirable when solving stiff ordinary differential equations or semi-discretized partial differential equations, as it allows larger time steps without numerical blow-up. However, A-stability is a linear concept and does not by itself guarantee good nonlinear behavior or accuracy. Other properties, such as L-stability and stiff accuracy, may be important depending on the problem. In practice, the choice of method balances stability, accuracy, and computational cost, with implicit, A-stable schemes forming a common core for stiff problems.