Scalar stability function, A-stability, and L-stability for Runge–Kutta methods

For a Runge–Kutta tableau $(A,b,c)$, apply its stage equations to the scalar test equation $y^{\mathrm{\prime }}=\lambda y$, with $z=h\lambda$. Whenever $I-zA$ is invertible, the stages satisfy $Y=(I-zA{)}^{-1}\mathbf{1}y$, and the scalar stability function is

$R(z)=1+z{b}^{\mathsf{T}}(I-zA{)}^{-1}\mathbf{1.}$

The scalar stability region is the set of $z\in \mathbb{C}$ for which the scalar update has $\mathrm{\mid }R(z)\mathrm{\mid }\le 1$. The method is A-stable when its rational stability function has no pole in $\{\mathrm{Re}z\le 0\}$ and $\mathrm{\mid }R(z)\mathrm{\mid }\le 1$ throughout that closed half-plane. It is L-stable when it is A-stable and $R(z)\to 0$ as $\mathrm{\mid }z\mathrm{\mid }\to \mathrm{\infty }$ with $\mathrm{Re}z\le 0$.

These are properties of the scalar linear test equation and the tableau's rational function. They do not assert nonlinear contractivity or control of nonnormal matrix growth.