Glossary

Region of absolute stability

The set of hλh\lambda values for which a method damps the test equation.

Definition

It is defined by applying the method to the test equation y′=λyy'=\lambda y: the region contains the complex values z=hλz=h\lambda for which the numerical solution stays bounded. Explicit Euler requires ∣1+z∣≤1|1+z|\le1; implicit Euler is stable on the whole half-plane Re⁡(z)≤0\operatorname{Re}(z)\le0 (A-stable).

How it is used

It translates stiffness into a concrete step condition: with real λ<0\lambda<0, explicit Euler needs h<2/∣λ∣h<2/|\lambda| even if accuracy asks for less.