Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 38C (Numerical Analysis)

Apply the numerical method to the test equation $y^{\prime }=\lambda y$ and put $z=h\lambda$. Its linear stability domain is the set of $z$ for which the computed amplification is stable, for a one-step method $|R(z)|\le 1$ with all stage solves well-defined. The method is A-stable if this includes the whole closed left half-plane (or its interior with boundary limits).

The Runge-Kutta method has $A=\left(\begin{array}{cc}1/4& 1/4-a\\ 1/4+a& 1/4\end{array}\right)$ and $b=(1/2,1/2{)}^T$. Solving the stage equations for $y^{\prime }=\lambda y$ gives the stability functionFor $a\ne 0$, the denominator roots have positive real parts: real roots are positive from their positive sum and product, while a complex conjugate pair has real part $1/(4a^2)>0$. For $a=0$ its sole root is $2$. Thus there are no poles or singular stage systems in the left half-plane.

Writing $z=x+iy$, direct expansion givesThis proves the A-stability of a symmetric two-stage implicit Runge-Kutta family: