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A-stability of a symmetric two-stage implicit Runge-Kutta family

For the two-stage implicit Runge-Kutta method with

A = (\frac{1}{4} \frac{1}{4} + a \frac{1}{4} - a \frac{1}{4}), b = (\frac{1}{2} \frac{1}{2}),

the stability function is

R (z) = \frac{2 + z + 2 a ^{2} z ^{2}}{2 - z + 2 a ^{2} z ^{2}} .

For

z = x + i y

,

∣2 - z + 2 a^{2} z^{2} ∣^{2} - ∣2 + z + 2 a^{2} z^{2} ∣^{2} = - 8 x (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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