Absolute Stability

Definition: A method is absolutely stable for a given step size and a given differential equation if the change due to a perturbation of size in one of the computed values is no larger than in all subsequent values

Remark: In contrast to the definition of stability, absolute stability is applied at a specific value of rather than in the limit as Also the definition of absolute stability depends heavily on the differential equation. In order to reduce this dependence, it is common to apply the concept to the “test equation”

(2.19)

where is a complex constant.

Definition: The region of absolute stability of a method is that set of all non-negative real values of and complex values of for which a perturbation in a single computed value will produce a change in subsequent values that does not increase from step to step, when applied to the test equation (2.19).