17A (Numerical Analysis)

Write the internal stage as

k_{2} = f (y_{n} + h [(1 - θ) k_{1} + θ k_{2}]) .

The two methods have

θ = 1/2

and

3/4

. Expansion gives stability function

R_{θ} (z) = \frac{1 + ( 1 - θ ) z + ( 1/2 - θ ) z ^{2}}{1 - θ z} = 1 + z + \frac{z ^{2}}{2} + \frac{θ}{2} z^{3} + O (z^{4}) .

Both therefore have order two; order three would require

θ = 1/3

b

{z \in C : ∣ R (z) ∣ \leq 1}

y^{'} = λ y

z = hλ

. A method is A-stable when this domain contains the entire closed left half-plane.

For

θ = 1/2

R (z) = \frac{1 + z /2}{1 - z /2},

∣ R (z) ∣ \leq 1

exactly when

Re z \leq 0

. Thus the first method is A-stable with stability domain the closed left half-plane. The

θ = 3/4

method is not A-stable: along the negative real axis its quadratic numerator makes

∣ R (z) ∣

grow without bound as

z \to - \infty

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