📂Numerical Analysis

A-Stable

Buildup

Multistep methods including the midpoint method might have parasitic solutions when $h$ is not sufficiently small. Being not sufficiently small refers to situations such as when there is a problem like $y ' = \lambda y$ and it fails to meet conditions like $| 1 + h \lambda| <1$.

When we say $z : = h \lambda \in \mathbb{C}$ and represent the condition on the complex plane, it looks like the figure below.

If $z$ does not belong to this region, the method does not work properly, and in cases where the magnitude is extremely large like $\lambda = - 10^{6}$, $h$ cannot be used unless it is substantially small. However, mindlessly reducing $h$ makes the computational cost too high to be practical.

Therefore, it’s preferable for such regions to be as large as possible, but with methods like the Adams-Moulton method, increasing the order makes the calculation more accurate but narrows the region guaranteed for stability.

Definition ¹

On the other hand, methods that guarantee stability for all $\operatorname{Re} ( h \lambda ) <0$ are called A-Stable. These methods do not impose any limitation on $h$, therefore they can solve many problems stably, which is an advantage.

Theorems

• [1]: There do not exist A-stable multistep methods of order higher than $2$.
• [2]: For every order of convergence, there exists an A-stable one-step method.

A-Stable methods are known for these two facts.