Area of ​​stability

In the numerics of ordinary differential equations , the stability domain of a method for solving an initial value problem is defined as the set of complex numbers with for which the numerical method is used in solving Dahlquist's test equation ${\displaystyle \xi =\lambda \Delta t}$ ${\displaystyle \lambda \in \mathbb {C} }$

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

provides a monotonically decreasing sequence of approximations with a fixed step size . This implies that the numerical method is stable for this equation and this step size . ${\displaystyle \Delta t}$

The case is particularly interesting when the stability region contains the complete left half-plane, then the method is called A-stable .

literature

• E. Hairer, G. Wanner: Solving Ordinary Differential Equations II, Stiff problems , Springer Verlag