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I'm trying to determine the stability region of the Heun method for ODEs by using the equation $y' = ky$, where $k$ is a complex number, based on the method described here.

If the Heun method is:

$$y_{n+1} = y_n + 0.5\cdot h\bigl(f(t_n, y_n) + f(t_{n+1},y_n + 0.5\cdot h\cdot f(t_n, y_n)\bigr)$$

then when I insert $y' = zy$ for $f(t,y)$, my result simplifies to

$$ y_{n+1} = (0.25\cdot h^2 \cdot z^2 + hz + 1)y_n $$

to judge from the wiki article, the stability region is then the area described by

$$\\{z \in \mathbb C \mid 0.25h^2z^2 + hz + 1 < 1\\}$$

Am I doing this right? What would such a region look like? Can someone help me get the intuition for this? And then I guess the method is A-stable if that region includes wherever $\Re < 0$?

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    Please, may I ask you?I want to know the stabilities such asbsolue stability, asymptitical stabiliy, A-stability and B - stability for numerical integration method.Thanks for this.2012-12-01

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