= A-stability of a symmetric two-stage implicit Runge-Kutta family
{c}
For the two-stage <implicit Runge-Kutta method> with
$$
A=
\begin{pmatrix}
\frac14&\frac14-a\\
\frac14+a&\frac14
\end{pmatrix},
\qquad
b=\begin{pmatrix}\frac12\\\frac12\end{pmatrix},
$$
the <stability function> is
$$
R(z)=\frac{2+z+2a^2z^2}{2-z+2a^2z^2}.
$$
For $z=x+iy$,
$$
|2-z+2a^2z^2|^2-|2+z+2a^2z^2|^2
=-8x(1+a^2|z|^2).
$$
The denominator has no zero in the closed left half-plane, so the method is <A-stable> for every real $a$.
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