# A stability - A-Stabilität

A numerical method for solving initial value problems is called A-stable if its stability domain contains the complete left half-plane of the complex number plane .

In other words, this means that the numerical method in solving the Dahlquist test equation

${\ displaystyle y '= \ lambda y, \ quad y (0) = 1}$

for all complex ${\displaystyle \lambda }$with negative real part at any step size${\displaystyle \Delta t}$yields a monotonically decreasing sequence of approximations . This implies that the method is stable regardless of the right hand side of the differential equation and does not develop oscillations. This property is important in solving rigid initial value problems .

Examples of A-stable methods are the implicit Euler method , the implicit trapezoid method , and BDF (2) .

The term was introduced by Germund Dahlquist in 1963 . He chose the A because attributes like "strong" or "super" seemed too trite to him. He also proved the second Dahlquist barrier , according to which A-stable linear multistep methods cannot be of order higher than 2. In contrast, implicit Runge-Kutta methods can also be A-stable at a higher order.

Explicit Runge-Kutta methods, just like explicit linear multi-step methods, always have a limited stability area, so they are never A-stable.

## L-stability

If one also demands that the stability function ${\displaystyle R(z)}$If the following equation is fulfilled, the procedure is called L-stable :

${\displaystyle \lim _{\operatorname {Re} (z)\to -\infty }|R(z)|=0.}$

This is relevant in order to be able to dampen oscillations quickly.

## variants

One method is called A (${\displaystyle \alpha }$) -stable if the area of ​​stability is the angle ${\displaystyle \alpha }$, starting from the zero point with the negative real axis as the bisector. So there are right-hand sides that cause problems for the process, but depending on the size of the angle, there are very few, and the time step is not limited for all others.

BDF processes are from order 3 to order 6 A (${\displaystyle \alpha }$) -stable, where the angle becomes smaller with higher order.

## literature

• G. Dahlquist: A Special Stability Problem for Linear Multistep Methods in BIT 3 (1), 27–43, 1963
• E. Hairer, G. Wanner: Solving Ordinary Differential Equations II, Stiff problems, Springer Verlag ISBN 3-540-60452-9